RD SHARMA Solutions for Class 12-science Maths Chapter 27 - Direction Cosines and Direction Ratios

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Chapter 27 - Direction Cosines and Direction Ratios Exercise Ex. 27.1

Question 1

If a line makes angles of 90°, 60° and 30° with the positive direction of x,y and z-axis respectively, find its direction cosines.

Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

 

begin mathsize 12px style text m end text equals text cos   60 end text degree equals 1 half end style


begin mathsize 12px style straight n equals cos text   end text 30 degree equals fraction numerator square root of 3 over denominator 2 end fraction end style

begin mathsize 12px style therefore straight T text he   direction   cosines   of   the   line   are   0 , end text 1 half comma fraction numerator square root of 3 over denominator 2 end fraction. end style

Question 2

If a line has direction ratios 2, -1, -2, determine its direction cosines.

Solution 2

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text the text    end text direction text    end text cosines text    end text of text    end text the text    end text line text    end text be text    end text straight l comma straight m comma straight n. end cell row cell Here comma end cell row cell straight a equals 2 comma straight b equals negative 1 comma straight c equals negative 2 text    end text are text    end text the text    end text direction text    end text ratios text    end text of text    end text the text    end text line. end cell row cell straight l equals plus-or-minus fraction numerator straight a over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight m equals plus-or-minus fraction numerator straight b over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction comma straight n equals plus-or-minus fraction numerator straight c over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 2 squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction end cell row cell straight l equals fraction numerator 2 over denominator square root of 9 end fraction comma straight m equals fraction numerator negative 1 over denominator square root of 9 end fraction comma straight n equals fraction numerator negative 2 over denominator square root of 9 end fraction end cell row cell straight l equals 2 over 3 comma straight m equals negative 1 third comma straight n equals negative 2 over 3 end cell row cell therefore 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 text    end text 2 over 3 comma negative 1 third comma negative 2 over 3. end cell end table end style

Question 3

Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).

Solution 3

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 joining text    end text left parenthesis negative 2 comma 4 comma negative 5 right parenthesis text   end text and text   end text left parenthesis 1 comma text   end text 2 comma text   end text 3 right parenthesis text   are end text comma end cell row cell left parenthesis 1 plus 2 comma 2 minus 4 comma 3 plus 5 right parenthesis equals left parenthesis 3 comma negative 2 comma 8 right parenthesis end cell row cell Here comma straight a equals 3 comma straight b equals negative 2 comma straight c equals 8 end cell row cell Direction text    end text cosines text    end text are end cell row cell fraction numerator 3 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator negative 2 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction comma fraction numerator 8 over denominator square root of 3 squared plus left parenthesis negative 2 right parenthesis squared plus 8 squared end root end fraction end cell row cell equals fraction numerator 3 over denominator square root of 77 end fraction comma fraction numerator negative 2 over denominator square root of 77 end fraction comma fraction numerator 8 over denominator square root of 77 end fraction end cell end table end style

Question 4

Using direction ratios show that the points A (2, 3, -4), B (1, -2, 3) and C (3, 8, -11) are collinear.

Solution 4

begin mathsize 12px style table attributes columnalign left end attributes row cell Here text    end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis. end cell row cell Direction text    end text ratios text    end text of text    end text AB equals left parenthesis 1 minus 2 comma negative 2 minus 3 comma 3 plus 4 right parenthesis equals left parenthesis negative 1 comma negative 5 comma 7 right parenthesis end cell row cell Direction text    end text ratios text    end text of text    end text BC equals left parenthesis 3 minus 1 comma 8 plus 2 comma negative 11 minus 3 right parenthesis equals left parenthesis 2 comma 10 comma negative 14 right parenthesis end cell row blank row cell Here comma text    end text the text    end text respective text    end text direction text    end text cosines text    end text of text    end text AB text    end text and text    end text AC comma end cell row cell fraction numerator negative 1 over denominator 2 end fraction equals fraction numerator negative 5 over denominator 10 end fraction equals fraction numerator 7 over denominator negative 14 end fraction text    end text are text    end text proportional. end cell row blank row cell Also comma text   end text straight B text    end text is text    end text the text    end text common text    end text point text    end text between text    end text the text    end text two text    end text lines comma end cell row cell therefore The text    end text points text     end text straight A text   end text left parenthesis 2 comma 3 comma negative text 4 end text right parenthesis comma text   end text straight B text   end text left parenthesis 1 comma negative 2 comma 3 right parenthesis text   end text and text   end text straight C text   end text left parenthesis 3 comma 8 comma negative 11 right parenthesis text    end text are text    end text collinear. text   end text end cell row blank end table end style

Question 5

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)

Solution 5

begin mathsize 12px style table attributes columnalign left end attributes row cell straight A left parenthesis 3 comma 5 comma negative 4 right parenthesis comma straight B left parenthesis negative 1 comma 1 comma 2 right parenthesis text    end text and text    end text straight C left parenthesis negative 5 comma negative 5 comma negative 2 right parenthesis end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AB equals left parenthesis negative 1 minus 3 comma 1 minus 5 comma 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 4 comma 6 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AB text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction comma fraction numerator 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus 6 squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator 6 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator 3 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text BC equals left parenthesis negative 5 plus 1 comma negative 5 minus 1 comma negative 2 minus 2 right parenthesis end cell row cell equals left parenthesis negative 4 comma negative 6 comma negative 4 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text BC text    end text will text    end text be end cell row cell fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 6 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction comma fraction numerator negative 4 over denominator square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 68 end fraction comma fraction numerator negative 6 over denominator square root of 68 end fraction comma fraction numerator negative 4 over denominator square root of 68 end fraction end cell row cell equals fraction numerator negative 2 over denominator square root of 17 end fraction comma fraction numerator negative 3 over denominator square root of 17 end fraction comma fraction numerator negative 2 over denominator square root of 17 end fraction end cell row cell The text    end text direction text    end text ratios text    end text of text    end text the text    end text side text    end text AC equals left parenthesis negative 5 minus 3 comma negative 5 minus 5 comma negative 2 plus 4 right parenthesis end cell row cell equals left parenthesis negative 8 comma negative 10 comma 2 right parenthesis end cell row cell Direction text    end text cosines text    end text of text    end text AC text    end text will text    end text be end cell row cell fraction numerator negative 8 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator negative 10 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction comma fraction numerator 2 over denominator square root of left parenthesis negative 8 right parenthesis squared plus left parenthesis negative 10 right parenthesis squared plus 2 squared end root end fraction end cell row cell equals fraction numerator negative 8 over denominator square root of 168 end fraction comma fraction numerator negative 10 over denominator square root of 168 end fraction comma fraction numerator 2 over denominator square root of 168 end fraction end cell row cell equals fraction numerator negative 4 over denominator square root of 42 end fraction comma fraction numerator negative 5 over denominator square root of 42 end fraction comma fraction numerator 1 over denominator square root of 42 end fraction end cell end table end style

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.

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(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Find the angle between the lines whose direction cosines are given by equations

2l + 2m - n = 0, mn + ln + lm = 0

Solution 16(iv)

  

Chapter 27 - Direction Cosines and Direction Ratios Exercise MCQ

Question 1

For every point P(x, y, z) on the xy-plane,

  1. x = 0
  2. y = 0
  3. z = 0
  4. x = y = z = 0
Solution 1

Correct option: (c)

Every point on xy-plane z co-ordinate is always zero.

 

Question 2

For every point P(x, y, z) on the x-axis (except the origin),

  1. x = 0, y = 0, z ≠ 0
  2. x = 0, z = 0, y ≠ 0
  3. y = 0, z = 0, x ≠ 0
  4. x = y = z = 0
Solution 2

Correct option: (c)

Point (x, y, z) is on the x-axis. Hence, y and z co-ordinate will be zero except the origin.

Question 3

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

 

  1. 2
  2. 3
  3. 4
  4. all of these
Solution 3

Correct option: (d)

Coordinates of the points given are diagonally opposite vertices of a parallelepiped. Hence, edges of parallelepiped can be 5-2, 7-3, 9-7 3,4,2.

Question 4

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

 

 

Solution 4

Correct option : (a)

  

 

Question 5

The xy-plane divides the line joining the points (-1, 3, 4) and (2, -5, 6)

 

  1. internally in the ratio 2 : 3
  2. externally in the ratio 2 : 3
  3. internally in the ratio 3 : 2
  4. externally in the ratio 3 : 2

 

Solution 5

Correct option: (b)

  

Question 6

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, -2) is 4, then its z-coordinate is

 

  1. 2
  2. 1
  3. -1
  4. -2
Solution 6

Correct option: (c)

  

Question 7

The distance of the point P(a, b, c) from the x-axis is

 

 

Solution 7

Correct option: (a)

  

Question 8

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

 

  1. 3 : 4 internally
  2. 3 : 1 externally
  3. 1 : 2 internally
  4. 2 : 1 externally

 

Solution 8

Correct option: (b)

  

Question 9

If P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear, then R divides PQ in the ratio

 

  1. 3 : 2 internally
  2. 3 : 2 externally
  3. 2 : 1 internally
  4. 2 : 1 externally

 

Solution 9

Correct option: (b)

  

 

Question 10

A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are the vertices of a triangle ABC. If the bisector of BAC meets BC at D, then coordinates of D are

  1. (19/8, 57/16, 17/16)
  2. (-19/8, 57/16, 17/16)
  3. (19/8, -57/16, 17/16)
  4. none of these
Solution 10

Correct option: (a)

  

Question 11

If O is the origin, OP = 3 with direction ratios proportional to -1, 2, -2 then the coordinates of P are

 

  1. (-1, 2, -2)
  2. (1, 2, 2)
  3. (-1/9, 2/9, -2/9)
  4. (3, 6, -9)
Solution 11

Correct option: (a)

Directions of OP from the origin

(-1,2,-2)=(x,y,z)

 

Question 12

The angle between the two diagonals of a cube is

 

Solution 12

Correct option: (d)

  

Question 13

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2α + cos2β + cos2γ + cos2δ is equal to

 

Solution 13

Correct option: (c)

  

 

Chapter 27 - Direction Cosines and Direction Ratios Exercise Ex. 27VSAQ

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

So the projection of the point P on y-axis is (0, -3, 0)

Question 16

Solution 16

Question 17

If a line has direction ratios proportional to 2, -1, -2, then what are its direction cosines?

Solution 17

Question 18

Write direction cosines of a line parallel to -axis.

Solution 18

Question 19

If a unit vector begin mathsize 12px style a with rightwards arrow on top end style makes an angle begin mathsize 12px style straight pi over 3 space with space straight i with hat on top comma space straight pi over 4 with space straight j with hat on top end style and an acute angle θ with k, then find the value of θ.

Solution 19

begin mathsize 12px style table attributes columnalign left end attributes row cell The text    end text direction text    end text cosines text    end text of text    end text the text    end text vector text    end text are end cell row cell straight l equals cos straight pi over 3 equals 1 half end cell row cell straight m equals cos straight pi over 4 equals fraction numerator 1 over denominator square root of 2 end fraction end cell row cell straight n equals cosθ end cell row blank row cell We text    end text know comma end cell row cell straight l squared plus straight m squared plus straight n squared equals 1 end cell row cell rightwards double arrow left parenthesis 1 half right parenthesis squared plus left parenthesis fraction numerator 1 over denominator square root of 2 end fraction right parenthesis squared plus cos squared straight theta equals 1 end cell row cell rightwards double arrow 1 fourth plus 1 half plus cos squared straight theta equals 1 end cell row cell rightwards double arrow cos squared straight theta equals 1 minus 1 half minus 1 fourth end cell row cell rightwards double arrow cos squared straight theta equals 1 fourth end cell row cell rightwards double arrow cosθ equals 1 half end cell row cell therefore straight theta equals straight pi over 3 end cell end table end style

Question 20

Write the distance of a point P (a, b, c) from x-axis.

Solution 20

begin mathsize 12px style table attributes columnalign left end attributes row cell Any text    end text point text    end text on text    end text the text    end text straight x minus axis text    end text can text    end text be text    end text written text    end text as text   end text left parenthesis straight a comma 0 comma 0 right parenthesis end cell row cell therefore Distance text    end text of text    end text the text    end text point text    end text from text    end text the text    end text straight x minus axis end cell row cell equals square root of left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 2 minus straight z subscript 1 right parenthesis squared end root end cell row cell equals square root of left parenthesis straight a minus straight a right parenthesis squared plus left parenthesis straight b minus 0 right parenthesis squared plus left parenthesis straight c minus 0 right parenthesis squared end root end cell row cell equals square root of 0 squared plus straight b squared plus straight c squared end root end cell row cell equals square root of straight b squared plus straight c squared end root end cell row cell therefore Distance text    end text of text    end text the text    end text point text    end text from text    end text the text    end text straight x minus axis text    end text is text    end text square root of straight b squared plus straight c squared end root. end cell end table end style