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

Scalar Triple Product Exercise Ex. 26.1

Solution 1(i)

Solution 1(ii)

Solution 2(i)

Solution 2(ii)

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

Solution 5(i)

Solution 5(ii)

Solution 5(iii)

Solution 5(iv)

Solution 6

Solution 7

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.

Solution 8

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

Solution 9

Solution 10

Solution 11

Solution 12(i)

Solution 12(ii)

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

Solution 13

Scalar Triple Product Exercise MCQ

Solution 1

Correct option: (a)

  

Solution 2

Correct option: (a)

  

Solution 3

Correct option: (a)

  

Solution 4

Correct option: (c)

  

Solution 5

Correct option: (d)

  

Solution 6

Correct option:(a)

  

Solution 7

Correct option: (c)

  

Solution 8

 

NOTE: Answer not matching with back answer.

Solution 9

Correct option: (a)

  

 

Solution 10

Correct option: (a)

  

Solution 11

Correct option: (d)

 

NOTE: Answer not matching with back answer.

 

Solution 12

Correct option: (c)

  

Solution 13

Correct option: (d)

  

 

Solution 14

Correct option: (d)

 

NOTE: Answer not matching with back answer.

Solution 15

Correct option: (c)

  

Scalar Triple Product Exercise Ex. 26VSAQ

Solution 1

Solution 2

Solution 3

begin mathsize 12px style left square bracket i with hat on top minus j with hat on top space space space space space j with hat on top minus k with hat on top space space space space space k with hat on top minus i with hat on top right square bracket end style
equals open square brackets size 12px i with size 12px hat on top space space space size 12px j with size 12px hat on top size 12px space size 12px minus size 12px k with size 12px hat on top size 12px space size 12px space size 12px space size 12px space size 12px space size 12px k with size 12px hat on top size 12px minus size 12px i with size 12px hat on top close square brackets minus begin mathsize 12px style left square bracket space j with hat on top space space space space space j with hat on top minus k with hat on top space space space space space k with hat on top minus i with hat on top right square bracket end style
equals begin mathsize 12px style open square brackets i with hat on top space space space space j with hat on top space space space k with hat on top minus i with hat on top close square brackets end style minus begin mathsize 12px style space open square brackets i with hat on top space space space k with hat on top space space space k with hat on top minus i with hat on top close square brackets end style minus begin mathsize 12px style open square brackets j with hat on top space space space space j with hat on top space space space space k with hat on top minus i with hat on top close square brackets end style plus begin mathsize 12px style left square bracket j with hat on top space space space k with hat on top space space space space k with hat on top minus i with hat on top right square bracket end style
size 12px equals begin mathsize 12px style left square bracket i with hat on top space space space j with hat on top space space space space k with hat on top right square bracket end style size 12px minus begin mathsize 12px style left square bracket i with hat on top space space space j with hat on top space space space space i with hat on top right square bracket end style size 12px minus begin mathsize 12px style left square bracket i with hat on top space space space k with hat on top space space space space k with hat on top right square bracket end style size 12px plus begin mathsize 12px style left square bracket i with hat on top space space space space k with hat on top space space space space i with hat on top right square bracket end style size 12px space size 12px minus size 12px 0 size 12px plus size 12px 0 size 12px minus begin mathsize 12px style left square bracket j with hat on top space space space k with hat on top space space space i with hat on top right square bracket end style
size 12px equals begin mathsize 12px style left square bracket i with hat on top space space space space space j with hat on top space space space space k with hat on top right square bracket end style size 12px minus size 12px 0 size 12px minus size 12px 0 size 12px plus size 12px 0 size 12px plus size 12px 0 size 12px minus size 12px 0 size 12px plus size 12px 0 size 12px minus begin mathsize 12px style left square bracket i with hat on top space space space space j with hat on top space space space space k with hat on top right square bracket end style
size 12px equals size 12px 0

Solution 4

begin mathsize 12px style open vertical bar table row 1 2 1 row straight alpha 1 2 row 1 2 straight alpha end table close vertical bar equals 0
1 left parenthesis straight alpha minus 4 right parenthesis minus 2 left parenthesis straight alpha squared minus 2 right parenthesis plus open parentheses 1 straight alpha minus 1 close parentheses equals 0
straight alpha minus 4 minus 2 straight alpha squared plus 4 plus 2 straight alpha minus 1 equals 0
2 straight alpha squared minus 3 straight alpha plus 1 equals 0
left parenthesis 2 straight alpha minus 1 right parenthesis left parenthesis straight alpha minus 1 right parenthesis equals 0
straight alpha equals 1 half comma 1 end style

Solution 5

Volume of the parallelepiped is given by

begin mathsize 12px style open vertical bar table row 1 1 0 row 1 2 0 row 1 1 straight pi end table close vertical bar equals 1 left parenthesis 2 straight pi minus 0 right parenthesis minus 1 left parenthesis straight pi minus 0 right parenthesis plus 0
space space space space space space space space space space space space space space space space equals 2 straight pi minus straight pi
space space space space space space space space space space space space space space space space equals straight pi end style

Thus, volume of the parallelopiped isbegin mathsize 12px style straight pi end style cubic units.

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

Solution 7

Solution 8

Solution 9

Solution 10

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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