RD SHARMA Solutions for Class 12-science Maths Chapter 26 - Scalar Triple Product

Your CBSE Class 12 syllabus for Maths consists of topics such as linear programming, vector quantities, determinants, etc. which lay the foundation for further education in science, engineering, management, etc. Studying differential equations will be useful for exploring subjects such as Physics, Biology, Chemistry, etc. where your knowledge can be applied for scientific investigations.

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Chapter 26 - Scalar Triple Product Exercise Ex. 26.1

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
Solution 16
Question 17

S h o w space t h a t space t h e space p o i n t s space A open parentheses minus 1 comma 4 comma minus 3 close parentheses comma space B open parentheses 3 comma 2 comma minus 5 close parentheses comma space C open parentheses minus 3 comma 8 comma minus 5 close parentheses space a n d space D open parentheses minus 3 comma 2 comma 1 close parentheses
a r e space c o minus p l a n a r.

Solution 17

A B equals p o s i t i o n space v e c t o r space o f space B minus p o s i t i o n space v e c t o r space o f space A
space space space space space equals 4 i with hat on top minus 2 j with hat on top minus 2 k with hat on top
A C equals p o s i t i o n space v e c t o r space o f space C minus p o s i t i o n space v e c t o r space o f space A
space space space space space equals minus 2 i with hat on top plus 4 j with hat on top minus 2 k with hat on top
A D equals p o s i t i o n space v e c t o r space o f space D minus p o s i t i o n space v e c t o r space o f space A
space space space space space equals minus 2 i with hat on top minus 2 j with hat on top plus 4 k with hat on top
T h e space f o u r space p o i n t s space a r e space c o minus p l a n a r space i f space t h e space v e c t o r s space a r e space c o minus p l a n a r.
T h u s comma space open vertical bar table row 4 cell minus 2 end cell cell minus 2 end cell row cell minus 2 end cell 4 cell minus 2 end cell row cell minus 2 end cell cell minus 2 end cell 4 end table close vertical bar equals 4 open square brackets 16 minus 4 close square brackets plus 2 open square brackets minus 8 minus 4 close square brackets minus 2 open square brackets 4 plus 8 close square brackets equals 48 minus 24 minus 24 equals 0
H e n c e space p r o v e d.

Question 18

S h o w space t h a t space t h e space f o l u r space p o i n t s space w h o s e space p o s i t i o n space v e c t o r s space a r e space 6 i with hat on top minus 7 j with hat on top comma
16 i with hat on top minus 19 j with hat on top minus 4 k with hat on top comma space 3 i with hat on top minus 6 k with hat on top comma space 2 i with hat on top minus 5 j with hat on top plus 10 k with hat on top

Solution 18

L e t space O A equals 6 i with hat on top minus 7 j with hat on top comma space O B equals 16 i with hat on top minus 19 j with hat on top minus 4 k with hat on top comma space O C equals 3 i with hat on top minus 6 k with hat on top comma space O D equals 2 i with hat on top minus 5 j with hat on top plus 10 k with hat on top
T h u s comma
A B equals O B minus O A equals 10 i with hat on top minus 12 j with hat on top minus 4 k with hat on top
A C equals O C minus O A equals minus 3 i with hat on top plus 7 j with hat on top minus 6 k with hat on top
A D equals O D minus O A equals minus 4 i with hat on top plus 2 j with hat on top plus 10 k with hat on top
T h e space f o u r space p o i n t s space a r e space c o minus p l a n a r space i f space v e c t o r s space A B comma space A C space a n d space A D space a r e space c o minus p l a n a r.
T h u s comma space w e space h a v e
open vertical bar table row 10 cell minus 12 end cell cell minus 4 end cell row cell minus 3 end cell 7 cell minus 6 end cell row cell minus 4 end cell 2 10 end table close vertical bar equals 10 open parentheses 70 plus 12 close parentheses plus 12 open parentheses minus 30 minus 24 close parentheses minus 4 open parentheses minus 6 plus 28 close parentheses equals 820 minus 648 minus 88

Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23

begin mathsize 12px style table attributes columnalign left end attributes row cell text let   end text straight a with rightwards arrow on top text  =  end text straight i with hat on top text + end text straight j with hat on top text + end text straight k with hat on top text ,  end text straight b with rightwards arrow on top text  =  end text straight i with hat on top text   and   end text stack text c end text with rightwards arrow on top text  =  end text straight c subscript 1 straight i with hat on top text + end text straight c subscript 2 straight j with hat on top text + end text straight c subscript 3 straight k with hat on top text .  end text end cell row cell text Then ,  If   end text straight c subscript text 2 end text end subscript text  = - 1   and   end text straight c subscript text 3 end text end subscript equals 1 comma text   show   that   no   value   end text end cell row cell text of   end text straight c subscript text 1 end text end subscript text   can   make   end text straight a with rightwards arrow on top comma text   end text straight b with rightwards arrow on top text   and   end text straight c with rightwards arrow on top text   coplanar. end text end cell end table end style

Solution 23

begin mathsize 12px style table attributes columnalign left end attributes row cell straight a with rightwards arrow on top text  =  end text straight i with hat on top text + end text straight j with hat on top text + end text straight k with hat on top text ,  end text straight b with rightwards arrow on top text  =  end text straight i with hat on top text   and   end text stack text c end text with rightwards arrow on top text  =  end text straight c subscript 1 straight i with hat on top text + end text straight c subscript 2 straight j with hat on top text + end text straight c subscript 3 straight k with hat on top text. end text end cell row cell straight c subscript text 2 end text end subscript text  = - 1   and   end text straight c subscript text 3 end text end subscript equals 1 comma end cell row cell text If   end text straight a with rightwards arrow on top comma straight b with rightwards arrow on top comma text    and   end text stack text c end text with rightwards arrow on top text   are   coplanar ,  then   their   scalar   triple   product   is   zero. end text end cell row cell left square bracket table row cell straight a with rightwards arrow on top end cell cell straight b with rightwards arrow on top end cell cell stack text c end text with rightwards arrow on top end cell end table right square bracket equals vertical line table row 1 1 1 row 1 0 0 row cell straight c subscript 1 end cell cell straight c subscript 2 end cell cell straight c subscript 3 end cell end table vertical line equals 0 end cell row cell rightwards double arrow negative straight c subscript 3 plus straight c subscript 2 equals 0 end cell row cell rightwards double arrow straight c subscript 2 equals straight c subscript 3 end cell row cell text But   this   is   a   contradiction   as   it   is   given   that   end text straight c subscript text 2 end text end subscript text  = - 1   and   end text straight c subscript text 3 end text end subscript equals 1. end cell row cell text Hence ,  no   value   of   end text straight c subscript 1 text   can   make   the   vectors   coplanar. end text end cell end table end style

Question 24
Solution 24

Chapter 26 - Scalar Triple Product Exercise MCQ

Question 1

Solution 1

Correct option: (a)

  

Question 2

Solution 2

Correct option: (a)

  

Question 3

Solution 3

Correct option: (a)

  

Question 4

Solution 4

Correct option: (c)

  

Question 5

Solution 5

Correct option: (d)

  

Question 6

Solution 6

Correct option:(a)

  

Question 7

Solution 7

Correct option: (c)

  

Question 8

Solution 8

 

NOTE: Answer not matching with back answer.

Question 9

Solution 9

Correct option: (a)

  

 

Question 10

Solution 10

Correct option: (a)

  

Question 11

Solution 11

Correct option: (d)

 

NOTE: Answer not matching with back answer.

 

Question 12

Solution 12

Correct option: (c)

  

Question 13

Solution 13

Correct option: (d)

  

 

Question 14

Solution 14

Correct option: (d)

 

NOTE: Answer not matching with back answer.

Question 15

Solution 15

Correct option: (c)

  

Chapter 26 - Scalar Triple Product Exercise Ex. 26VSAQ

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Thus, volume of the parallelopiped is straight pi cubic units.

Question 6

I f space a comma space b space a r e space n o n minus c o l l i n e a r space v e c t o r s comma space t h e n space f i n d space t h e space v a l u e space o f space
open square brackets a with rightwards arrow on top space b with rightwards arrow on top space i with hat on top close square brackets i with hat on top plus open square brackets a with rightwards arrow on top space b with rightwards arrow on top space j with hat on top close square brackets j with hat on top plus open square brackets a with rightwards arrow on top space b with rightwards arrow on top space k with hat on top close square brackets k with hat on top

Solution 6

F o r space a n y space v e c t o r space r with rightwards arrow on top comma space w e space h a v e
r with rightwards arrow on top equals open parentheses stack r. with rightwards arrow on top i with overparenthesis on top close parentheses i with overparenthesis on top plus open parentheses stack r. with rightwards arrow on top j with overparenthesis on top close parentheses j with overparenthesis on top plus open parentheses stack r. with rightwards arrow on top k with overparenthesis on top close parentheses k with overparenthesis on top
R e p l a c i n g space r with rightwards arrow on top space b y space a with rightwards arrow on top cross times b with rightwards arrow on top comma space w e space h a v e
a with rightwards arrow on top cross times b with rightwards arrow on top equals open vertical bar open parentheses a with rightwards arrow on top cross times b with rightwards arrow on top close parentheses times i with overparenthesis on top close vertical bar i with overparenthesis on top plus open vertical bar open parentheses a with rightwards arrow on top cross times b with rightwards arrow on top close parentheses times j with overparenthesis on top close vertical bar j with overparenthesis on top plus open vertical bar open parentheses a with rightwards arrow on top cross times b with rightwards arrow on top close parentheses times k with overparenthesis on top close vertical bar k with overparenthesis on top
rightwards double arrow a with rightwards arrow on top cross times b with rightwards arrow on top equals open square brackets a with rightwards arrow on top space b with rightwards arrow on top space i with overparenthesis on top close square brackets i with overparenthesis on top plus open square brackets a with rightwards arrow on top space b with rightwards arrow on top space j with overparenthesis on top close square brackets j with overparenthesis on top plus open square brackets a with rightwards arrow on top space b with rightwards arrow on top space k with overparenthesis on top close square brackets k with overparenthesis on top

Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11

begin mathsize 12px style text Find   end text a with rightwards arrow on top. left parenthesis b with rightwards arrow on top cross times c with rightwards arrow on top right parenthesis comma text   if   end text a with rightwards arrow on top equals 2 i with hat on top plus j with hat on top plus 3 k with hat on top comma b with rightwards arrow on top equals negative i with hat on top plus 2 j with hat on top plus k with hat on top text   and   end text c with rightwards arrow on top equals 3 i with hat on top plus j with hat on top plus 2 k with hat on top. end style

Solution 11

begin mathsize 12px style table attributes columnalign left end attributes row cell a with rightwards arrow on top equals 2 i with hat on top plus j with hat on top plus 3 k with hat on top comma b with rightwards arrow on top equals negative i with hat on top plus 2 j with hat on top plus k with hat on top text   and   end text c with rightwards arrow on top equals 3 i with hat on top plus j with hat on top plus 2 k with hat on top end cell row cell b with rightwards arrow on top cross times c with rightwards arrow on top equals vertical line table row cell i with hat on top end cell cell j with hat on top end cell cell k with hat on top end cell row cell negative 1 end cell 2 1 row 3 1 2 end table vertical line equals i with hat on top left parenthesis 4 minus 1 right parenthesis minus j with hat on top left parenthesis negative 5 right parenthesis plus k with hat on top left parenthesis negative 1 minus 6 right parenthesis end cell row cell equals 3 i with hat on top plus 5 j with hat on top minus 7 k with hat on top end cell row cell a with rightwards arrow on top. left parenthesis b with rightwards arrow on top cross times c with rightwards arrow on top right parenthesis equals left parenthesis 2 i with hat on top plus j with hat on top plus 3 k with hat on top right parenthesis. left parenthesis 3 i with hat on top plus 5 j with hat on top minus 7 k with hat on top right parenthesis end cell row cell equals 6 plus 5 minus 21 equals negative 10 end cell end table end style

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