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Let a=3hat(i) +2hat(j)+xhat(k) " and " ...

Let `a=3hat(i) +2hat(j)+xhat(k) " and " b=hat(i)-hat(j) +hat(k) ` for some real x Then `|axx b| =r` is possible if

A

`0 lt r le sqrt((3)/(2))`

B

`sqrt((3)/(2)) lt r le 3 sqrt((3)/(2))`

C

`3sqrt((3)/(2)) lt r lt 5sqrt((3)/(2))`

D

`r ge 5 sqrt((3)/(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
Given vectors are ` a = 3hat(i) + 2hat(j) + xhat(k)`
` " and " b= hat(i) - hat(j) + hat(k)`
`:. , a xx b = |{:(hat(i) ,,hat(j) ,,hat(k) ),(3,,2,,1),(1,,-1,,1):}|= hat(i) (2 +x) - hat(j) (3-x) + hat(k) (-3-2)`
`= (x+2)hat(i) + (x-3) hat(j) = 5hat(k)`
` rArr | axx b | = sqrt( (x+2)^(2) + (x-3)^(2) + 25)`
`=sqrt(2x^(2) -2x+ 4+9 +25)`
`=sqrt(2(x^(2)-x +(1)/(4)) -(1)/(2) + 38) = sqrt(2(x-(1)/(2))^(2) + (75)/(2))`
`= sqrt(2(x^(2) -x+ (1)/(4)) - (1)/(2) + 38 )= sqrt(2(x-(1)/(2))^(2) + (75)/(2))`
`So | axx b| ge sqrt(75)/(2)) [ "at " x=(1)/(2) ,| a xx b| " is minimum "]`
`rArr r ge 5 sqrt((3)/(2))`
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