The value of `lambda,` for which the four points `2hati+3hatj-hatk, hati+3hatk, 2hati+4hatj-2hatk, hati-6hatj+lambda hatk` are coplanar, is

Options

A. 2
B. 4
C. 6
D. 8

Answer

C

Explanation

Let `A=2hat(i)+3hat(j)-hat(k), B =hat(i)-2hat(j)+3hat(k)`,
`C=3hat(i)+4hat(j)-2hat(k)` and `D=hat(i)-6hat(j)+lambda hat(k)`
Now, `AB=-hat(i)-5hat(j)+4hat(k), AC =hat(i)+hat(j)-hat(k)`
and `AD=-hat(i)-9hat(j)+(lambda+1)hat(k)`
These will be coplanar, if `[(AB,AC,AD)]=0`
`:. |(-1,-5,4),(1,1,-1),(-1,-9,(lambda+1))|=0`
`implies -1(lambda+1-9)+5(lambda+1-1)+4(-9+1)=0`
`implies 4lambda -24 =0`
`implies lambda =6`