The position vectors of the points A,B, C and D are `3hati - 2hatj - hatk,2hati + 3hatj - 34hatk, -hati +hatj + 2hatk` and `4hati + 5hatj + lambdahatk` respectively. If the points A, B ,C and D lie on a plane, find the value of `lambda`.

Answer

`-(146)/(17)`

Explanation

Given that P.V.'s of points A, B, C and D are `3hati -2hatj -hatk, 2hati +3hatj -4hatk, -hati +hatj +2hatk and 4hati +5hatj +lamda hatk`, respectively.
Given that A, B, C and D lie in a plane if
`vec(AB), vec(AC) and vec(AD)` are coplanar. Therefore,
`|{:(-1,,5,,-3),(-4,,3,,3),(1,,7,,1+lamda):}| =0`
or `-1(3+3lamda - 21)- 5(-4-4lamda -3)-3(-28 -3)=0`
or `-3lamda + 18 + 20 lamda + 35 + 93 =0`
or ` 17 lamda = -146`
or `lamda = - (146)/(17)`