Let `a= hat(i) + 2hat(j) + 4hat(k) " " b= hat(i) + lambda hat(j) +4hat(k)` and `c= 2hat(i) +4hat(j)+(lambda^(2)-1) hat(k)` be coplanar vectors . Then the non-zero vectors `axx c` is

Options

A. `-10 hat(i) +5hat(j)`
B. `-10hat(i)-5hat(j)`
C. `-14hat(i)-5hat(j)`
D. `-14hat(i)+5hat(j)`

Answer

A

Explanation

We know that if a,b,c are coplanar vectors then `[a,b,c]=0`
`:. |{:(1,,2,,4),(1,,lambda,,4),(2,,4,,lambda^(2)-1):}|=0`
`rArr 1{lambda(lambda^(2)-1) -16} -2(lambda^(2) -1) -8) +4(4-2lambda) =0`
`rArr lambda^(2) -lambda -16 -2lambda^(2) + 18 +16 -8lambda =0`
`rArr lambda^(2) -2lambda^(2) -9lambda +18 =0`
`rArr lambda^(2) (lambda-2) -9 (lambda - 2) = 0 `
`rArr (lambda -2 ) (lambda^(2) -9) =0`
`rArr (lambda - 2) (lambda+3) (lambda-3 =0`
`:. lambda=2 , 3 " or " -3` lt brgt If `lambda =2 ` then
`a xx c = |{:(hat(i) ,,hat(j),,hat(k) ),(1,,2,,4),(2,,4,,3):}|=hat(i)(6-16)-hat(j) (3-8) + hat(k) (4-4)`
`=- 10hat(i) + 5hat(j)`
If `lambda = +- 3 " then " a xx c|{:(hat(i) ,,hat(j) ,,hat(k)),(1,,2,,4),(2,,4,,3):}|=0`
(because last two rows are proportional)