RD SHARMA Solutions for Class 12-science Maths Chapter 28 - Straight line in space

Chapter 28 - Straight line in space Exercise Ex. 28.1

Question 1

begin mathsize 12px style table attributes columnalign left end attributes row cell text Find   the   vector   and   Cartesian   equations   of   the   line   end text end cell row cell text through   the   point  ( 5 ,  2 , - 4 )  and   which   is   parallel   end text end cell row cell text to   the   vector   3 end text straight i with hat on top plus 2 straight j with hat on top minus 8 straight k with hat on top. end cell end table end style

Solution 1

begin mathsize 12px style table attributes columnalign left end attributes row cell Vector text   end text equation text   end text of text   end text straight a text   end text line end cell row cell is text    end text straight r with rightwards arrow on top equals straight a with rightwards arrow on top plus straight lambda straight b with rightwards arrow on top text   end text end cell row cell The text    end text Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text is end cell row cell fraction numerator straight x minus straight x subscript 1 over denominator straight a subscript 1 end fraction equals fraction numerator straight y minus straight y subscript 1 over denominator straight a subscript 2 end fraction equals fraction numerator straight x minus straight x subscript 3 over denominator straight a subscript 3 end fraction end cell row cell Using text    end text the text    end text above text    end text formula comma end cell row cell Vector text    end text equation text    end text of text    end text the text    end text line comma end cell row cell straight r with rightwards arrow on top equals left parenthesis 5 straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top right parenthesis plus straight lambda left parenthesis text 3 end text straight i with hat on top plus 2 straight j with hat on top minus 8 straight k with hat on top right parenthesis end cell row cell The text    end text Cartesian text    end text equation text    end text of text    end text the text    end text line end cell row cell fraction numerator straight x minus 5 over denominator 3 end fraction equals fraction numerator straight y minus 2 over denominator 2 end fraction equals fraction numerator straight z plus 4 over denominator negative 8 end fraction end cell end table end style

Question 2

Find the vector equation of the line passing through the points (-1, 0, 2) and (3, 4, 6).

Solution 2

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text are end cell row cell left parenthesis 3 plus 1 comma 4 minus 0 comma 6 minus 2 right parenthesis equals left parenthesis 4 comma 4 comma 4 right parenthesis end cell row cell Since text    end text the text    end text line text    end text passes text    end text through text    end text left parenthesis negative 1 comma 0 comma 2 right parenthesis end cell row cell The text    end text vector text    end text equation text    end text of text    end text the text    end text line comma end cell row cell straight r with rightwards arrow on top equals straight a with rightwards arrow on top plus straight lambda straight b with rightwards arrow on top end cell row cell rightwards double arrow straight r with rightwards arrow on top equals straight a with rightwards arrow on top plus straight lambda straight b with rightwards arrow on top end cell row cell rightwards double arrow straight r with rightwards arrow on top equals left parenthesis negative straight i with rightwards arrow on top plus 0 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top right parenthesis plus straight lambda left parenthesis 4 straight i with rightwards arrow on top plus 4 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top right parenthesis end cell row cell therefore The text    end text vector text    end text equation text    end text of text    end text the text    end text line comma end cell row cell straight r with rightwards arrow on top equals left parenthesis negative straight i with rightwards arrow on top plus 0 straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top right parenthesis plus straight lambda left parenthesis 4 straight i with rightwards arrow on top plus 4 straight j with rightwards arrow on top plus 4 straight k with rightwards arrow on top right parenthesis end cell end table end style

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

.

Question 17

Solution 17

Chapter 28 - Straight line in space Exercise Ex. 28.2

Question 1

begin mathsize 12px style table attributes columnalign left end attributes row cell text Show   that   the   three   lines   with   direction   cosines   end text fraction numerator text 12 end text over denominator text 13 end text end fraction comma fraction numerator negative 3 over denominator 13 end fraction comma fraction numerator negative 4 over denominator 13 end fraction semicolon end cell row cell 4 over 13 comma 12 over 13 comma 3 over 13 semicolon 3 over 13 comma fraction numerator negative 4 over denominator 13 end fraction comma 12 over 13 text   are   mutually   perpendicular. end text end cell end table end style

Solution 1

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text straight l subscript 1 equals 12 over 13 comma straight m subscript 1 equals negative 3 over 13 comma straight n subscript 1 equals negative 4 over 13 end cell row cell straight l subscript 2 equals 4 over 13 comma straight m subscript 2 equals 12 over 13 comma straight n subscript 1 equals 3 over 13 end cell row cell straight l subscript 3 equals 3 over 13 comma straight m subscript 3 equals negative 4 over 13 comma straight n subscript 3 equals 12 over 13 end cell row blank row cell straight l subscript 1 straight l subscript 2 plus straight m subscript 1 straight m subscript 2 plus straight n subscript 1 straight n subscript 2 end cell row cell equals 12 over 13 cross times 4 over 13 plus left parenthesis negative 3 over 13 right parenthesis cross times 12 over 13 plus left parenthesis negative 4 over 13 right parenthesis cross times 3 over 13 end cell row cell equals fraction numerator 48 minus 36 minus 12 over denominator 169 end fraction equals 0 end cell row cell straight l subscript 2 straight l subscript 3 plus straight m subscript 2 straight m subscript 3 plus straight n subscript 2 straight n subscript 3 end cell row cell equals 4 over 13 cross times 3 over 13 plus 12 over 13 cross times left parenthesis negative 4 over 13 right parenthesis plus 3 over 13 cross times 12 over 13 end cell row cell equals fraction numerator 12 minus 48 plus 36 over denominator 169 end fraction equals 0 end cell row cell straight l subscript 1 straight l subscript 3 plus straight m subscript 1 straight m subscript 3 plus straight n subscript 1 straight n subscript 3 end cell row cell equals 12 over 13 cross times 3 over 13 plus left parenthesis negative 3 over 13 right parenthesis cross times left parenthesis negative 4 over 13 right parenthesis plus left parenthesis negative 4 over 13 right parenthesis cross times 12 over 13 end cell row cell equals fraction numerator 36 plus 12 minus 48 over denominator 169 end fraction equals 0 end cell row cell therefore The text    end text lines text    end text are text    end text mutually text    end text perpendicular. end cell end table end style

Question 2

Show that the line through the points (1, -1, 2) and (3, 4, -2) is perpendicular to the line through points (0, 3, 2) and (3, 5, 6).

Solution 2

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text   ratios   of    end text straight a text    end text line text    end text passing text    end text through text    end text the text    end text points end cell row cell left parenthesis 1 comma negative 1 comma 2 right parenthesis text    end text and text   end text left parenthesis 3 comma 4 comma negative 2 right parenthesis text    end text are end cell row cell left parenthesis 3 minus 1 comma 4 plus 1 comma negative 2 minus 2 right parenthesis end cell row cell equals left parenthesis 2 comma 5 comma negative 4 right parenthesis end cell row blank row cell The text    end text direction text   ratios    end text of text    end text straight a text    end text line text    end text passing text    end text through text    end text the text    end text points end cell row cell left parenthesis 0 comma 3 comma 2 right parenthesis text    end text and text   end text left parenthesis 3 comma 5 comma 6 right parenthesis text    end text are end cell row cell left parenthesis 3 minus 0 comma 5 minus 3 comma 6 minus 2 right parenthesis end cell row cell equals left parenthesis 3 comma 2 comma 4 right parenthesis end cell row blank row cell Angle text    end text between text    end text the text    end text lines text    end text end cell row cell cosθ equals fraction numerator left parenthesis straight a subscript 1 straight a subscript 2 plus straight b subscript 1 straight b subscript 2 plus straight c subscript 1 straight c subscript 2 right parenthesis over denominator square root of straight a subscript 1 squared plus straight b subscript 1 squared plus straight c subscript 1 squared end root square root of straight a subscript 2 squared plus straight b subscript 2 squared plus straight c subscript 2 squared end root end fraction end cell row cell cosθ equals fraction numerator left square bracket 2 cross times 3 plus 5 cross times 2 plus left parenthesis negative 4 right parenthesis cross times 4 right square bracket over denominator square root of 2 squared plus 5 squared plus left parenthesis negative 4 right parenthesis squared end root square root of 3 squared plus 2 squared plus 4 squared end root end fraction end cell row cell cosθ equals fraction numerator 0 over denominator square root of 2 squared plus 5 squared plus left parenthesis negative 4 right parenthesis squared end root square root of 3 squared plus 2 squared plus 4 squared end root end fraction end cell row cell cosθ equals 0 end cell row cell straight theta equals straight pi over 2 end cell row cell The text    end text lines text    end text are text    end text mutually text    end text perpendicular. end cell end table end style

Question 3

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (-1, -2, 1) and (1, 2, 5).

Solution 3

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text   ratios    end text of text    end text straight a text    end text line text    end text passing text    end text through text    end text the text    end text points end cell row cell left parenthesis 4 comma 7 comma 8 right parenthesis text    end text and text   end text left parenthesis 2 comma 3 comma 4 right parenthesis text    end text are end cell row cell left parenthesis 4 minus 2 comma 7 minus 3 comma 8 minus 4 right parenthesis end cell row cell equals left parenthesis 2 comma 4 comma 4 right parenthesis end cell row cell The text    end text direction text   ratios    end text of text    end text straight a text    end text line text    end text passing text    end text through text    end text the text    end text points end cell row cell left parenthesis negative 1 comma negative 2 comma 1 right parenthesis text    end text and text   end text left parenthesis 1 comma 2 comma 5 right parenthesis text    end text are end cell row cell left parenthesis negative 1 minus 1 comma negative 2 minus 2 comma 1 minus 5 right parenthesis end cell row cell equals left parenthesis negative 2 comma negative 4 comma negative 4 right parenthesis end cell row blank row cell The text    end text direction text   ratios   are   proportional. end text end cell row cell fraction numerator text 2 end text over denominator negative 2 end fraction equals fraction numerator 4 over denominator negative 4 end fraction equals fraction numerator 4 over denominator negative 4 end fraction end cell row cell Hence comma text   end text the text    end text lines text    end text are text    end text mutually text    end text parallel. end cell end table end style

Question 4

begin mathsize 12px style table attributes columnalign left end attributes row cell text Find   the   Cartesian   equation   of   the   line   which   passes   end text end cell row cell text through   the   point  (- 2 ,  4 , - 5 )  and   parallel   to   the   line   end text end cell row cell text given   by   end text fraction numerator text x + 3 end text over denominator text 3 end text end fraction equals fraction numerator straight y minus 4 over denominator 5 end fraction equals fraction numerator straight z plus 8 over denominator 6 end fraction end cell end table end style

Solution 4

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text passing text    end text through text    end text left parenthesis straight x subscript 1 comma straight y subscript 1 comma straight z subscript 1 right parenthesis end cell row cell and text    end text with text   end text direction text    end text ratios text   end text left parenthesis straight a subscript 1 comma straight b subscript 1 comma straight c subscript 1 right parenthesis end cell row cell fraction numerator straight x minus straight x subscript 1 over denominator straight a subscript 1 end fraction equals fraction numerator straight y minus straight y subscript 1 over denominator straight b subscript 1 end fraction equals fraction numerator straight z minus straight z subscript 1 over denominator straight c subscript 1 end fraction end cell row cell The text    end text Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text passing text    end text through text   end text left parenthesis negative 2 comma 4 comma negative 5 right parenthesis end cell row cell and text    end text parallel text    end text to text    end text the text    end text line text     end text fraction numerator text x + 3 end text over denominator text 3 end text end fraction equals fraction numerator straight y minus 4 over denominator 5 end fraction equals fraction numerator straight z plus 8 over denominator 6 end fraction text     end text is end cell row cell fraction numerator straight x plus 2 over denominator 3 end fraction equals fraction numerator straight y minus 4 over denominator 5 end fraction equals fraction numerator straight z plus 5 over denominator 6 end fraction end cell end table end style

Question 5

  

Solution 5

  

Question 6

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1) and (4, 3, -1).

Solution 6

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text ratios text    end text of text   end text straight a text   end text line text    end text joining text    end text the text    end text origin text    end text to text    end text the text    end text point left parenthesis 2 comma 1 comma 1 right parenthesis end cell row cell are text    end text left parenthesis 2 minus 0 comma 1 minus 0 comma 1 minus 0 right parenthesis equals left parenthesis 2 comma 1 comma 1 right parenthesis end cell row cell The text    end text direction text    end text ratios text    end text of text   end text straight a text   end text line text    end text joining text    end text left parenthesis 3 comma 5 comma negative 1 right parenthesis text    end text and text    end text left parenthesis 4 comma 3 comma negative 1 right parenthesis end cell row cell are text    end text left parenthesis 4 minus 3 comma 3 minus 5 comma negative 1 plus 1 right parenthesis equals left parenthesis 1 comma negative 2 comma 0 right parenthesis end cell row blank row cell Angle text     end text between text    end text the text    end text lines end cell row cell cos text   end text straight theta equals fraction numerator straight a subscript 1 straight a subscript 2 plus straight b subscript 1 straight b subscript 2 plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 squared plus straight b subscript 1 squared plus straight c subscript 1 squared end root square root of straight a subscript 2 squared plus straight b subscript 2 squared plus straight c subscript 2 squared end root end fraction end cell row cell cos text   end text straight theta equals fraction numerator 2 cross times 1 plus 1 cross times left parenthesis negative 2 right parenthesis plus 1 cross times 0 over denominator square root of 2 squared plus 1 squared plus 1 squared end root square root of 1 squared plus left parenthesis negative 2 right parenthesis squared plus 0 squared end root end fraction end cell row cell cos text   end text straight theta equals fraction numerator 0 over denominator square root of 6 square root of 5 end fraction end cell row cell cos text   end text straight theta equals 0 end cell row cell therefore straight theta equals straight pi over 2 end cell row cell The text    end text lines text    end text are text    end text mutually text    end text perpendicular. end cell end table end style

Question 7

Find the equation of a line parallel to x-axis and passing through the origin.

Solution 7

begin mathsize 12px style table attributes columnalign left end attributes row cell Vector text    end text equation text    end text of text    end text straight a text    end text line text    end text is text    end text end cell row cell straight r with rightwards arrow on top equals straight a with rightwards arrow on top plus straight lambda straight b with rightwards arrow on top end cell row cell The text    end text direction text    end text cosines text    end text of text    end text the text   end text straight x minus axis text    end text are text    end text left parenthesis 1 comma 0 comma 0 right parenthesis. text    end text Equation text    end text of text    end text straight a text    end text line text    end text parallel text   end text end cell row cell text   end text to text    end text the text    end text straight x minus axis text    end text and text    end text passing text    end text through text    end text the text    end text origin text    end text is text    end text end cell row cell straight r with rightwards arrow on top equals left parenthesis 0 straight i with hat on top plus 0 straight j with hat on top plus 0 straight k with hat on top right parenthesis plus straight lambda left parenthesis 1 straight i with hat on top plus 0 straight j with hat on top plus 0 straight k with hat on top right parenthesis end cell row cell straight r with rightwards arrow on top equals straight lambda straight i with hat on top end cell end table end style

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 9(v)

Solution 9(v)

Question 9(vi)

find the angle between the following pairs of line :

fraction numerator minus x plus 2 over denominator minus 2 end fraction equals fraction numerator y minus 1 over denominator 7 end fraction equals fraction numerator z plus 3 over denominator minus 3 end fraction space a n d space fraction numerator x plus 2 over denominator minus 1 end fraction equals fraction numerator 2 y minus 8 over denominator 4 end fraction equals fraction numerator z minus 5 over denominator 4 end fraction

Solution 9(vi)

Question 10(i)

Solution 10(i)

Question 10(ii)

Solution 10(ii)

Question 10(iii)

Solution 10(iii)

Question 10(iv)

find the angle between the pairs of lines with directions ratios proposal to a, b, c and b - c, c - a, a - b.

Solution 10(iv)

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

  

Solution 21

  

Question 22

Solution 22

Question 23

Solution 23

Question 24

begin mathsize 12px style table attributes columnalign left end attributes row cell text Find   the   direction   cosines   of   the   line   end text fraction numerator x plus 2 over denominator 2 end fraction equals fraction numerator 2 y minus 7 over denominator 6 end fraction equals fraction numerator 5 minus z over denominator 6 end fraction. end cell row cell text Also ,  find   the   vector   equation   of   the   line   through   the   point   end text end cell row cell text A (- 1 ,  2 ,  3 )  and   parallel   to   the   given   line. end text end cell end table end style

Solution 24

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text line text    end text are end cell row cell fraction numerator straight x plus 2 over denominator 2 end fraction equals fraction numerator 2 straight y minus 7 over denominator 6 end fraction equals fraction numerator 5 minus straight z over denominator 6 end fraction end cell row cell 2 comma 6 comma 6 end cell row blank row cell The text    end text direction text    end text cosines text    end text of text    end text the text    end text line text    end text are end cell row cell straight l equals fraction numerator 2 over denominator square root of 2 squared plus 6 squared plus 6 squared end root end fraction equals fraction numerator 2 over denominator square root of 76 end fraction end cell row cell straight m equals fraction numerator 6 over denominator square root of 2 squared plus 6 squared plus 6 squared end root end fraction equals fraction numerator 6 over denominator square root of 76 end fraction end cell row cell straight n equals fraction numerator 6 over denominator square root of 2 squared plus 6 squared plus 6 squared end root end fraction equals fraction numerator 6 over denominator square root of 76 end fraction end cell row cell left parenthesis fraction numerator 2 over denominator square root of 76 end fraction comma fraction numerator 6 over denominator square root of 76 end fraction comma fraction numerator 6 over denominator square root of 76 end fraction right parenthesis end cell row blank row cell therefore Vector text    end text equation text    end text of text    end text the text    end text line text    end text is end cell row cell straight r with rightwards arrow on top equals straight a with rightwards arrow on top plus straight lambda straight b with rightwards arrow on top end cell row cell straight r with rightwards arrow on top equals left parenthesis negative straight i with rightwards arrow on top plus 2 straight j with rightwards arrow on top plus 3 straight k with rightwards arrow on top right parenthesis plus straight lambda left parenthesis 2 straight i with rightwards arrow on top plus 6 straight j with rightwards arrow on top plus 6 straight k with rightwards arrow on top right parenthesis end cell end table end style

Chapter 28 - Straight line in space Exercise Ex. 28.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6(i)

Solution 6(i)

Question 6(ii)

Solution 6(ii)

Question 6(iii)

Solution 6(iii)

Question 6(iv)

Solution 6(iv)

Question 7

begin mathsize 12px style table attributes columnalign left end attributes row cell text Show   that   the   lines   end text stack text r end text with rightwards arrow on top equals 3 straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top plus straight lambda left parenthesis straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis text   and   end text end cell row cell stack text r end text with rightwards arrow on top equals 5 straight i with hat on top minus 2 straight j with hat on top plus straight mu left parenthesis 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top right parenthesis text   are   intersecting .  Hence , end text end cell row cell text find   their   point   of   intersection. end text end cell end table end style

Solution 7

begin mathsize 12px style table attributes columnalign left end attributes row cell stack text r end text with rightwards arrow on top equals 3 straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top plus straight lambda left parenthesis straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis end cell row cell stack text r end text with rightwards arrow on top equals 5 straight i with hat on top minus 2 straight j with hat on top plus straight mu left parenthesis 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top right parenthesis end cell row cell If text    end text the text    end text lines text    end text intersect text    end text each text    end text other comma text    end text then text    end text the text    end text shortest end cell row cell distance text    end text between text    end text the text    end text lines text    end text should text    end text be text    end text zero. end cell row cell Here comma end cell row cell stack straight a subscript 1 with rightwards arrow on top equals 3 straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top end cell row cell stack straight a subscript 2 with rightwards arrow on top equals 5 straight i with hat on top minus 2 straight j with hat on top end cell row cell stack straight b subscript 1 with rightwards arrow on top equals straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top end cell row cell stack straight b subscript 2 with rightwards arrow on top equals 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top end cell row cell left parenthesis stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top right parenthesis equals vertical line table row cell straight i with rightwards arrow on top end cell cell straight j with rightwards arrow on top end cell cell straight k with rightwards arrow on top end cell row 1 2 2 row 3 2 6 end table vertical line end cell row cell equals straight i with rightwards arrow on top left parenthesis 12 minus 4 right parenthesis minus straight j with rightwards arrow on top left parenthesis 6 minus 6 right parenthesis plus straight k with rightwards arrow on top left parenthesis 2 minus 6 right parenthesis end cell row cell equals 8 straight i with rightwards arrow on top minus 0 straight j with rightwards arrow on top minus 4 straight k with rightwards arrow on top end cell row cell left parenthesis stack straight a subscript 2 with rightwards arrow on top minus stack straight a subscript 1 with rightwards arrow on top right parenthesis equals left parenthesis 5 straight i with hat on top minus 2 straight j with hat on top minus 3 straight i with hat on top minus 2 straight j with hat on top plus 4 straight k with hat on top right parenthesis equals left parenthesis 2 straight i with hat on top minus 4 straight j with hat on top plus 4 straight k with hat on top right parenthesis end cell row cell Shortest text    end text Distance comma straight d equals vertical line fraction numerator left parenthesis stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top right parenthesis. left parenthesis stack straight a subscript 2 with rightwards arrow on top minus stack straight a subscript 1 with rightwards arrow on top right parenthesis over denominator vertical line stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top vertical line end fraction vertical line end cell row cell equals vertical line fraction numerator left parenthesis 8 straight i with rightwards arrow on top minus 0 straight j with rightwards arrow on top minus 4 straight k with rightwards arrow on top right parenthesis. left parenthesis 2 straight i with hat on top minus 4 straight j with hat on top plus 4 straight k with hat on top right parenthesis over denominator vertical line 8 straight i with rightwards arrow on top minus 0 straight j with rightwards arrow on top minus 4 straight k with rightwards arrow on top vertical line end fraction vertical line end cell row cell equals vertical line fraction numerator 8 cross times 2 minus 0 cross times 4 plus left parenthesis negative 4 right parenthesis cross times 4 over denominator vertical line 8 straight i with rightwards arrow on top minus 0 straight j with rightwards arrow on top minus 4 straight k with rightwards arrow on top vertical line end fraction vertical line end cell row cell equals vertical line fraction numerator 0 over denominator vertical line 8 straight i with rightwards arrow on top minus 0 straight j with rightwards arrow on top minus 4 straight k with rightwards arrow on top vertical line end fraction vertical line equals 0 end cell row cell Since text    end text the text    end text shortest text    end text distance text    end text is text    end text zero comma text    end text the text    end text lines end cell row cell are text    end text intersect text    end text each text    end text other. end cell end table end style

 

begin mathsize 12px style table attributes columnalign left end attributes row cell Point text    end text of text    end text intersection text    end text of text    end text the text    end text lines comma end cell row cell stack text r end text with rightwards arrow on top equals 3 straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top plus straight lambda left parenthesis straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis end cell row cell stack text r end text with rightwards arrow on top equals 5 straight i with hat on top minus 2 straight j with hat on top plus straight mu left parenthesis 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top right parenthesis end cell row cell Lines text    end text in text    end text the text    end text Cartesian text    end text form comma end cell row cell fraction numerator straight x minus 3 over denominator 1 end fraction equals fraction numerator straight y minus 2 over denominator 2 end fraction equals fraction numerator straight z plus 4 over denominator 2 end fraction equals straight lambda end cell row cell straight x equals straight lambda plus 3 comma straight y equals 2 straight lambda plus 2 comma straight z equals 2 straight lambda minus 4 end cell row cell fraction numerator straight x minus 5 over denominator 3 end fraction equals fraction numerator straight y plus 2 over denominator 2 end fraction equals straight z over 6 equals straight mu end cell row cell straight x equals 3 straight mu plus 5 comma straight y equals 2 straight mu minus 2 comma straight z equals 6 straight mu end cell end table end style

 

begin mathsize 12px style table attributes columnalign left end attributes row cell From text    end text coordinates text    end text of text    end text straight x comma end cell row cell straight lambda plus 3 equals 3 straight mu plus 5 end cell row cell straight lambda equals 3 straight mu plus 2..... left parenthesis straight i right parenthesis end cell row cell From text    end text coordinates text    end text of text    end text straight y comma end cell row cell 2 straight lambda plus 2 equals 2 straight mu minus 2 end cell row cell straight lambda equals straight mu minus 2....... left parenthesis ii right parenthesis end cell row cell Solving text    end text left parenthesis straight i right parenthesis text    end text and text    end text left parenthesis ii right parenthesis comma end cell row cell straight lambda equals negative 4 comma straight mu equals negative 2 end cell row blank row cell Coordinates text    end text of text    end text the text    end text point text    end text of text    end text intersection comma end cell row cell straight x equals 3 left parenthesis negative 2 right parenthesis plus 5 comma straight y equals 2 left parenthesis negative 2 right parenthesis minus 2 comma straight z equals 6 left parenthesis negative 2 right parenthesis end cell row cell straight x equals negative 1 comma straight y equals negative 6 comma straight z equals negative 12 end cell row cell left parenthesis negative 1 comma negative 6 comma negative 12 right parenthesis end cell end table end style

Chapter 28 - Straight line in space Exercise Ex. 28.4

Question 1

Solution 1


Question 2

Solution 2

Question 3

Solution 3

.

.
Question 4

Solution 4

Question 5

Find the foot of perpendicular from the point (2, 3, 4) to the line begin mathsize 12px style fraction numerator 4 minus straight x over denominator 2 end fraction equals straight y over 6 equals fraction numerator 1 minus straight z over denominator 3 end fraction end style. Also find the perpendicular distance from the given point to the line.

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

  

Solution 13

  

Question 14

Find the coordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, -1, 3) and C (2, -3, -1).

Solution 14

  

Chapter 28 - Straight line in space Exercise Ex. 28.5

Question 1(i)

Find the shortest distance between the pair of lines whose vector equation are

begin mathsize 12px style space space space space space space space space space space space r with rightwards arrow on top equals 3 i with hat on top space plus space 8 j with hat on top space plus space 3 k with hat on top space plus space lambda open parentheses 3 i with hat on top space minus space j with hat on top space plus space k with hat on top close parentheses
and comma space space r with rightwards arrow on top equals negative 3 i with hat on top space minus space 7 j with hat on top space plus space 6 k with hat on top space plus space mu open parentheses negative 3 i with hat on top space plus space 2 j with hat on top space plus space 4 k with hat on top close parentheses end style

Solution 1(i)

Question 1(ii)

Find the shortest distance between the pair of lines whose vector equations are:

begin mathsize 12px style space space space space space space space space space space space r with rightwards arrow on top equals 3 i with hat on top space plus space 5 j with hat on top space plus space 7 k with hat on top space plus space lambda open parentheses i with hat on top space minus space 2 j with hat on top space plus 7 space k with hat on top close parentheses
and comma space space r with rightwards arrow on top equals negative i with hat on top space minus space j with hat on top space minus space 6 k with hat on top space plus space mu open parentheses 7 i with hat on top space minus space 6 j with hat on top space plus space k with hat on top close parentheses end style

Solution 1(ii)

Question 1(iii)

Find the shortest distance between the pair of lines whose vector equations are:

begin mathsize 12px style space space space space space space space space space space space r with rightwards arrow on top equals open parentheses i with hat on top space plus space 2 j with hat on top space plus space 3 k with hat on top close parentheses space plus space lambda open parentheses 2 i with hat on top space plus 3 j with hat on top space plus space 4 k with hat on top close parentheses
and comma space space r with rightwards arrow on top equals open parentheses 2 i with hat on top space plus space 4 j with hat on top space plus space 5 k with hat on top close parentheses space plus space mu open parentheses 3 i with hat on top space plus space 4 j with hat on top space plus space 5 k with hat on top close parentheses end style

Solution 1(iii)

Question 1(iv)

Find the shortest distance between the lines whose vector equations are:

begin mathsize 12px style r with rightwards arrow on top equals open parentheses 1 minus t close parentheses stack i space with hat on top space plus space open parentheses t space minus space 2 close parentheses j with hat on top space plus space open parentheses 3 space minus space t close parentheses k with hat on top space a n d
r with rightwards arrow on top equals open parentheses s plus 1 close parentheses i with hat on top space plus space open parentheses 2 s minus 1 close parentheses j with hat on top space minus space open parentheses 2 s plus 1 close parentheses k with hat on top end style

Solution 1(iv)

Question 1(v)

Find the shortest distance between the pair of lines whose vector equations are:

begin mathsize 12px style space space space space space space space space space space space r with rightwards arrow on top equals open parentheses lambda space minus space 1 close parentheses i with hat on top space plus space open parentheses lambda space plus space 1 close parentheses j with hat on top space minus space space left parenthesis 1 space plus space lambda right parenthesis k with hat on top
and comma space space r with rightwards arrow on top equals left parenthesis 1 space minus space mu right parenthesis i with hat on top space plus space left parenthesis 2 mu space minus space 1 right parenthesis j with hat on top space plus space space left parenthesis mu space plus space 2 right parenthesis k with hat on top end style

Solution 1(v)

Question 1(vi)

Find the shortest distance between the pair of lines whose vector equations are:

begin mathsize 12px style r with rightwards arrow on top equals open parentheses 2 i with hat on top space minus space j with hat on top space minus space k with hat on top close parentheses space plus space lambda open parentheses 2 i with hat on top space minus space 5 j with hat on top space plus space 2 k with hat on top close parentheses semicolon space r with rightwards arrow on top equals open parentheses i with hat on top space plus space 2 j with hat on top space plus space k with hat on top close parentheses space plus space mu open parentheses i with hat on top space minus space j with hat on top space plus space k with hat on top close parentheses end style

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

  

Solution 1(viii)

  

Question 2(i)

Find the shortest distance between the pair of lines whose cartesian equations are:

begin mathsize 12px style fraction numerator straight x minus 1 over denominator 2 end fraction equals fraction numerator straight y minus 2 over denominator 3 end fraction equals fraction numerator straight z minus 3 over denominator 4 end fraction space and space fraction numerator straight x minus 2 over denominator 3 end fraction equals fraction numerator straight y minus 3 over denominator 4 end fraction equals fraction numerator straight z minus 5 over denominator 5 end fraction end style.

Solution 2(i)

Question 2(ii)

Find the shortest distance between the pair of lines whose cartesian equations are:

begin mathsize 12px style fraction numerator straight x minus 1 over denominator 2 end fraction equals fraction numerator straight y plus 1 over denominator 3 end fraction equals straight z space and space fraction numerator straight x plus 1 over denominator 3 end fraction equals fraction numerator straight y minus 2 over denominator 1 end fraction semicolon space straight z equals 2 end style.

Solution 2(ii)

Question 2(iii)

Find the shortest distance between the pair of lines whose cartesian equations are:

begin mathsize 12px style fraction numerator straight x minus 1 over denominator negative 1 end fraction equals fraction numerator straight y plus 2 over denominator 1 end fraction equals fraction numerator straight z minus 3 over denominator negative 2 end fraction and fraction numerator straight x minus 1 over denominator 1 end fraction equals fraction numerator straight y plus 1 over denominator 2 end fraction equals fraction numerator straight z plus 1 over denominator negative 2 end fraction end style.

Solution 2(iii)

Question 2(iv)

Find the shortest distance between the pair of lines whose Cartesian equations are:

begin mathsize 12px style fraction numerator x minus 3 over denominator 1 end fraction equals fraction numerator y minus 5 over denominator negative 2 end fraction equals fraction numerator z minus 7 over denominator 1 end fraction semicolon fraction numerator x plus 1 over denominator 7 end fraction equals fraction numerator y plus 1 over denominator negative 6 end fraction equals fraction numerator z plus 1 over denominator 1 end fraction end style.

Solution 2(iv)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Find the shortest distance between the lines

begin mathsize 12px style stack text r end text with rightwards arrow on top equals left parenthesis straight i with rightwards arrow on top plus 2 straight j with rightwards arrow on top plus straight k with rightwards arrow on top right parenthesis plus straight lambda left parenthesis straight i with rightwards arrow on top minus straight j with rightwards arrow on top plus straight k with rightwards arrow on top right parenthesis text   and   end text stack text r end text with rightwards arrow on top equals 2 straight i with rightwards arrow on top minus straight j with rightwards arrow on top minus straight k with rightwards arrow on top plus straight mu left parenthesis 2 straight i with rightwards arrow on top plus straight j with rightwards arrow on top plus 2 straight k with rightwards arrow on top right parenthesis end style

Solution 7(i)

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma end cell row cell straight a subscript 1 equals straight i with hat on top plus 2 straight j with hat on top plus straight k with hat on top end cell row cell straight b subscript 1 equals straight i with hat on top minus straight j with hat on top plus straight k with hat on top end cell row cell straight a subscript 2 equals 2 straight i with hat on top minus straight j with hat on top minus straight k with hat on top end cell row cell straight b subscript 2 equals 2 straight i with hat on top plus straight j with hat on top plus 2 straight k with hat on top end cell row blank row cell straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript blank subscript 1 end subscript equals 2 straight i with hat on top minus straight j with hat on top minus straight k with hat on top minus straight i with hat on top minus 2 straight j with hat on top minus straight k with hat on top equals straight i with hat on top minus 3 straight j with hat on top minus 2 straight k with hat on top end cell row cell straight b subscript 1 cross times straight b subscript 2 equals open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 1 end cell 1 row 2 1 2 end table close vertical bar equals straight i with hat on top left parenthesis negative 2 minus 1 right parenthesis minus straight j with hat on top left parenthesis 2 minus 2 right parenthesis plus straight k with hat on top left parenthesis 1 plus 2 right parenthesis equals negative 3 straight i with hat on top plus 3 straight k with hat on top end cell row blank row cell The text    end text shortest text    end text distance text    end text between text    end text the text    end text two text    end text lines comma end cell row cell straight d equals open vertical bar fraction numerator left parenthesis straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 right parenthesis. left parenthesis straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript 1 right parenthesis over denominator vertical line straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 vertical line end fraction close vertical bar end cell row cell straight d equals open vertical bar fraction numerator left parenthesis negative 3 straight i with hat on top plus 3 straight k with hat on top right parenthesis. left parenthesis straight i with hat on top minus 3 straight j with hat on top minus 2 straight k with hat on top right parenthesis over denominator vertical line minus 3 straight i with hat on top minus 3 straight k with hat on top vertical line end fraction close vertical bar equals open vertical bar fraction numerator negative 3 minus 6 over denominator square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis negative 3 right parenthesis squared end root end fraction close vertical bar equals fraction numerator 9 over denominator 3 square root of 2 end fraction end cell row cell The text    end text shortest text    end text distance text    end text between text    end text the text    end text two text    end text lines equals fraction numerator 3 over denominator square root of 2 end fraction units end cell end table end style

Question 7(ii)

Find the shortest distance between the lines


begin mathsize 12px style fraction numerator straight x plus 1 over denominator 7 end fraction equals fraction numerator straight y plus 1 over denominator negative 6 end fraction equals fraction numerator straight z plus 1 over denominator 1 end fraction text   and   end text fraction numerator text x end text minus text 3 end text over denominator text 1 end text end fraction equals fraction numerator straight y minus 5 over denominator negative 2 end fraction equals fraction numerator straight z minus 7 over denominator 1 end fraction end style

Solution 7(ii)

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma end cell row cell straight a with rightwards arrow on top subscript 1 equals negative straight i with hat on top minus straight j with hat on top minus straight k with hat on top end cell row cell straight a with rightwards arrow on top subscript 2 equals 3 straight i with hat on top plus 5 straight j with hat on top plus 7 straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 1 equals 7 straight i with hat on top minus 6 straight j with hat on top plus straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 2 equals straight i with hat on top minus 2 straight j with hat on top plus straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 equals open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 7 cell negative 6 end cell 1 row 1 cell negative 2 end cell 1 end table close vertical bar end cell row cell equals straight i with hat on top left parenthesis negative 6 plus 2 right parenthesis minus straight j with hat on top left parenthesis 7 minus 1 right parenthesis plus straight k with hat on top left parenthesis negative 14 plus 6 right parenthesis end cell row cell equals negative 4 straight i with hat on top minus 6 straight j with hat on top minus 8 straight k with hat on top end cell row cell straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript 1 equals straight i with hat on top left parenthesis 3 plus 1 right parenthesis plus straight j with hat on top left parenthesis 5 plus 1 right parenthesis plus straight k with hat on top left parenthesis 7 plus 1 right parenthesis end cell row cell equals 4 straight i with hat on top plus 6 straight j with hat on top plus 8 straight k with hat on top end cell row blank row cell The text    end text shortest text    end text distance text    end text between text    end text two text    end text lines comma end cell row cell straight d equals open vertical bar fraction numerator left parenthesis straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 right parenthesis. left parenthesis straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript 1 right parenthesis over denominator vertical line straight b subscript 1 cross times straight b subscript 2 vertical line end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator left parenthesis negative 4 straight i with hat on top minus 6 straight j with hat on top minus 8 straight k with hat on top right parenthesis. left parenthesis 4 straight i with hat on top plus 6 straight j with hat on top plus 8 straight k with hat on top right parenthesis over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 8 right parenthesis squared end root end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator negative 16 minus 36 minus 64 over denominator square root of 116 end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator negative 116 over denominator square root of 116 end fraction close vertical bar end cell row cell equals 2 square root of 29 text   end text units end cell end table end style

Question 7(iii)

Find the shortest distance between the lines

 

begin mathsize 12px style stack text r end text with rightwards arrow on top equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top plus straight lambda left parenthesis straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top right parenthesis text and   end text stack text r end text with rightwards arrow on top equals 4 straight i with hat on top plus 5 straight j with hat on top plus 6 straight k with hat on top plus straight mu left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus straight k with hat on top right parenthesis end style

Solution 7(iii)

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma end cell row cell straight a with rightwards arrow on top subscript 1 equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top comma end cell row cell straight b with rightwards arrow on top subscript 1 equals straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top end cell row cell straight a with rightwards arrow on top subscript 2 equals 4 straight i with hat on top plus 5 straight j with hat on top plus 6 straight k with hat on top comma end cell row cell straight b with rightwards arrow on top subscript 2 equals 2 straight i with hat on top plus 3 straight j with hat on top plus straight k with hat on top end cell row blank row cell left parenthesis straight a with rightwards arrow on top subscript blank subscript 2 end subscript minus straight a with rightwards arrow on top subscript blank subscript 1 end subscript right parenthesis equals 4 straight i with hat on top plus 5 straight j with hat on top plus 6 straight k with hat on top minus straight i with hat on top minus 2 straight j with hat on top minus 3 straight k with hat on top equals 3 straight i with hat on top plus 3 straight j with hat on top plus 3 straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 end cell row cell equals left parenthesis straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top right parenthesis cross times left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus straight k with hat on top right parenthesis end cell row cell equals open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 3 end cell 2 row 2 3 1 end table close vertical bar end cell row cell equals straight i with hat on top left parenthesis negative 3 minus 6 right parenthesis minus straight j with hat on top left parenthesis 1 minus 4 right parenthesis plus straight k with hat on top left parenthesis 3 plus 6 right parenthesis end cell row cell equals negative 9 straight i with hat on top plus 3 straight j with hat on top plus 9 straight k with hat on top end cell row cell Shortest text    end text distance text    end text between text    end text the text    end text two text    end text lines equals open vertical bar fraction numerator left parenthesis straight a subscript 2 minus straight a subscript 1 right parenthesis. left parenthesis straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 right parenthesis over denominator vertical line straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 vertical line end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator left parenthesis 3 straight i with hat on top plus 3 straight j with hat on top plus 3 straight k with hat on top right parenthesis. left parenthesis negative 9 straight i with hat on top plus 3 straight j with hat on top plus 9 straight k with hat on top right parenthesis over denominator vertical line minus 9 straight i with hat on top plus 3 straight j with hat on top plus 9 straight k with hat on top vertical line end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator 3 cross times left parenthesis negative 9 right parenthesis plus 3 cross times 3 plus 3 cross times 9 over denominator square root of left parenthesis negative 9 right parenthesis squared plus 3 squared plus 9 squared end root end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator negative 27 plus 9 plus 27 over denominator square root of left parenthesis negative 9 right parenthesis squared plus 3 squared plus 9 squared end root end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator 9 over denominator square root of 171 end fraction close vertical bar equals fraction numerator 3 over denominator square root of 19 end fraction units end cell end table end style

Question 7(iv)

Find the shortest distance between the lines

 

begin mathsize 12px style stack text r end text with rightwards arrow on top equals 6 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top plus straight lambda left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis text   and   end text stack text r end text with rightwards arrow on top equals negative 4 straight i with hat on top minus straight k with hat on top plus straight mu left parenthesis 3 straight i with hat on top minus 2 straight j with hat on top minus 2 straight k with hat on top right parenthesis end style

Solution 7(iv)

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma end cell row cell straight a with rightwards arrow on top subscript 1 equals 6 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top end cell row cell straight a with rightwards arrow on top subscript 2 equals negative 4 straight i with hat on top minus straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 1 equals straight i with hat on top minus 2 straight j with hat on top plus 2 straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 2 equals 3 straight i with hat on top minus 2 straight j with hat on top minus 2 straight k with hat on top end cell row blank row cell straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript 1 equals negative 4 straight i with hat on top minus straight k with hat on top minus 6 straight i with hat on top minus 2 straight j with hat on top minus 2 straight k with hat on top end cell row cell equals negative 10 straight i with hat on top minus 2 straight j with hat on top minus 3 straight k with hat on top end cell row cell straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 equals open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 2 end cell 2 row 3 cell negative 2 end cell cell negative 2 end cell end table close vertical bar end cell row cell equals straight i with hat on top left parenthesis 4 plus 4 right parenthesis minus straight j with hat on top left parenthesis negative 2 minus 6 right parenthesis plus straight k with hat on top left parenthesis negative 2 plus 6 right parenthesis end cell row cell equals 8 straight i with hat on top plus 8 straight j with hat on top plus 4 straight k with hat on top end cell end table end style

 

begin mathsize 12px style table attributes columnalign left end attributes row cell Shortest text    end text Distance equals open vertical bar fraction numerator left parenthesis straight a with rightwards arrow on top subscript 2 minus straight a with rightwards arrow on top subscript 1 right parenthesis. left parenthesis straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 right parenthesis over denominator vertical line straight b with rightwards arrow on top subscript 1 cross times straight b with rightwards arrow on top subscript 2 vertical line end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator left parenthesis negative 10 straight i with hat on top minus 2 straight j with hat on top minus 3 straight k with hat on top right parenthesis. left parenthesis 8 straight i with hat on top plus 8 straight j with hat on top plus 4 straight k with hat on top right parenthesis over denominator vertical line 8 straight i with hat on top minus 8 straight j with hat on top plus 4 straight k with hat on top vertical line end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator left parenthesis negative 10 right parenthesis cross times 8 plus left parenthesis negative 2 right parenthesis cross times 8 plus left parenthesis negative 3 right parenthesis cross times 4 over denominator square root of 8 squared plus left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction close vertical bar end cell row cell equals open vertical bar fraction numerator negative 80 minus 16 minus 12 over denominator square root of 64 plus 64 plus 16 end root end fraction close vertical bar equals open vertical bar fraction numerator negative 108 over denominator square root of 144 end fraction close vertical bar equals 9 text   units end text end cell end table end style

Question 8

Find the distance between the lines l1 and l2 given by

 

begin mathsize 12px style table attributes columnalign left end attributes row cell stack text r end text with rightwards arrow on top equals straight i with hat on top plus 2 straight j with hat on top minus 4 straight k with hat on top plus straight lambda left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis text   and   end text end cell row cell stack text r end text with rightwards arrow on top equals 3 straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top plus straight mu left parenthesis 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis end cell end table end style

Solution 8

  

  

Chapter 28 - Straight line in space Exercise MCQ

Question 1

  1. 45°
  2. 30°
  3. 60°
  4. 90°
Solution 1

Correct option: (d)

  

Question 2

  1. coincident
  2. skew
  3. intersecting
  4. parallel
Solution 2

Correct option: (a)

  

Question 3

  1. 4, 5, 7
  2. 4, -5, 7
  3. 4, -5, -7
  4. -4, 5, 7
Solution 3

Correct option: (a)

  

Question 4

 

Solution 4

Correct option: (c)

  

Question 5

Solution 5

Correct option: (a)

  

Question 6

  1. 7
  2. 5
  3. 0
  4. none of these
Solution 6

Correct option: (a)

  

 

Question 7

Solution 7

Correct option: (c)

  

Question 8

If a line makes angles α, β and γ with the axes respectively, then cos 2 α + cos 2 β + cos 2 γ =

  1. -2
  2. -1
  3. 1
  4. 2
Solution 8

Correct option: (b)

 

NOTE: Answer not matching with back answer.

Question 9

If the direction ratios of a line are proportional to 1, -3, 2, then its direction cosines are

Solution 9

Correct option: (a)

  

Question 10

Solution 10

Correct option: (b)

  

Question 11

The projections of a line segment on X, Y and Z axes are 12, 4 and 3 respectively. The length and direction cosines of the line segment are

Solution 11

Correct option: (a)

  

Question 12

  1. parallel
  2. intersecting
  3. skew
  4. coincident
Solution 12

Correct option: (d)

NOTE: Answer not matching with back answer. 

Question 13

  1. parallel to x-axis
  2. parallel to y-axis
  3. parallel to z-axis
  4. perpendicular to z-axis
Solution 13

Correct option: (d)

  

Question 14

Solution 14

Correct option: (d)

  

Chapter 28 - Straight line in space Exercise Ex. 28VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

If the equations of a line AB are  begin mathsize 12px style fraction numerator 3 minus straight x over denominator 2 end fraction equals fraction numerator straight y plus 2 over denominator negative 2 end fraction equals fraction numerator straight z minus 5 over denominator 4 end fraction end style, write the direction ratios of a line parallel to AB.

Solution 16

Question 17

Write the vector equaion of a line given by begin mathsize 12px style fraction numerator straight x minus 5 over denominator 3 end fraction equals fraction numerator straight y plus 4 over denominator 2 end fraction equals fraction numerator straight z minus 6 over denominator 2 end fraction end style

Solution 17

Question 18

.

Solution 18

Question 19

begin mathsize 12px style table attributes columnalign left end attributes row cell text Find   the   cartesian   equations   of   the   line   which   passes end text end cell row cell text through   the   point  (- 2 ,  4 , - 5 )  and   is   parallel   to   the   line end text end cell row cell fraction numerator straight x plus 3 over denominator 3 end fraction equals fraction numerator 4 minus straight y over denominator 5 end fraction equals fraction numerator straight z plus 8 over denominator 6 end fraction. end cell end table end style

Solution 19

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text is comma end cell row cell fraction numerator straight x plus 3 over denominator 3 end fraction equals fraction numerator 4 minus straight y over denominator 5 end fraction equals fraction numerator straight z plus 8 over denominator 6 end fraction end cell row cell fraction numerator straight x plus 3 over denominator 3 end fraction equals fraction numerator straight y minus 4 over denominator negative 5 end fraction equals fraction numerator straight z plus 8 over denominator 6 end fraction end cell row blank row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text given text    end text line end cell row cell are text    end text 3 comma negative 5 comma 6. end cell row cell Since text    end text the text    end text lines text    end text are text    end text parallel comma text    end text the text    end text direction text    end text ratios text    end text of text    end text the text    end text required text    end text line end cell row cell are text    end text 3 comma negative 5 comma 6. end cell row blank row cell Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text is comma end cell row cell fraction numerator straight x minus straight x subscript 1 over denominator straight a end fraction equals fraction numerator straight y minus straight y subscript 1 over denominator straight b end fraction equals fraction numerator straight z minus straight z subscript 1 over denominator straight c end fraction end cell row blank row cell The text    end text Cartesian text    end text equation text    end text of text    end text straight a text    end text line text    end text passing text    end text through end cell row cell left parenthesis negative 2 comma 4 comma negative 5 right parenthesis text    end text is end cell row cell fraction numerator straight x plus 2 over denominator 3 end fraction equals fraction numerator straight y minus 4 over denominator negative 5 end fraction equals fraction numerator straight z plus 5 over denominator 6 end fraction end cell end table end style

Question 20

begin mathsize 12px style table attributes columnalign left end attributes row cell text Find   the   angle   between   the   lines   end text end cell row cell stack text r end text with rightwards arrow on top equals left parenthesis 2 straight i with hat on top minus 5 straight j with hat on top plus straight k with hat on top right parenthesis plus straight lambda left parenthesis 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top right parenthesis end cell row cell text and   end text stack text r end text with rightwards arrow on top equals 7 straight i with hat on top minus 6 straight k with hat on top plus straight mu left parenthesis straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis. end cell end table end style

Solution 20

begin mathsize 12px style table attributes columnalign left end attributes row cell stack text r end text with rightwards arrow on top equals left parenthesis 2 straight i with hat on top minus 5 straight j with hat on top plus straight k with hat on top right parenthesis plus straight lambda left parenthesis 3 straight i with hat on top plus 2 straight j with hat on top plus 6 straight k with hat on top right parenthesis end cell row cell stack text r end text with rightwards arrow on top equals 7 straight i with hat on top minus 6 straight k with hat on top plus straight mu left parenthesis straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top right parenthesis end cell row blank row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text first text   end text line text    end text are text    end text straight a subscript 1 equals 3 comma straight b subscript 1 equals 2 comma straight c subscript 1 equals 6. end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text second text   end text line text    end text are text    end text straight a subscript 2 equals 1 comma straight b subscript 2 equals 2 comma straight c subscript 2 equals 2. end cell row blank row cell The text    end text angle text    end text between text    end text the text    end text two text    end text lines comma end cell row cell cosθ equals fraction numerator straight a subscript 1 straight a subscript 2 plus straight b subscript 1 straight b subscript 2 plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a squared plus straight b squared plus straight c squared end root square root of straight a squared plus straight b squared plus straight c squared end root end fraction end cell row cell equals fraction numerator 3 cross times 1 plus 2 cross times 2 plus 6 cross times 2 over denominator square root of 3 squared plus 2 squared plus 6 squared end root square root of 1 squared plus 2 squared plus 2 squared end root end fraction end cell row cell equals fraction numerator 3 plus 4 plus 12 over denominator square root of 49 square root of 9 end fraction end cell row cell equals fraction numerator 19 over denominator 7 cross times 3 end fraction end cell row cell equals 19 over 21 end cell row cell straight theta equals cos to the power of negative 1 end exponent left parenthesis 19 over 21 right parenthesis end cell end table end style