Find the value of `lambda` so that the four points with position vectors
`(-6hat(i)+3hat(j)+2hat(k)), (3 hat(i)+lambda hat(j)+4hat(k)), (5hat(i)+7hat(j)+3hat(k))and (-13hat(i)+17hat(j)-2hat(k))` are coplanar.

Explanation

Let the given points be A, B, C, D respectively. Then,
`vec(AB)`=(p.v. of B)-(p.v. of A)
`=(3hat(i)+lambdahat(j)+ 4 hat (k))-(-6hat(i)+3hat(j)+2hat(k))`
`=9hat(i)+(lambda-3)hat(j)+2hat(k)`
`vec(AC)`=(p.v. of C)-(p.v. of A)
`=(5hat(i)+7hat(j)+3hat(k))-(-6hat(i)+3hat(j)+2hat(k))=(11hat(i)+4hat(j)+hat(k))`
`vec(AD)=`(p.v. of D)-(p.v. of A)
`=(-13hat(i)+17hat(j)-hat(k))-(-6hat(i)+3hat(j)+2hat(k))`
`=(-7hat(i)+14hat(j)-3hat(k))`
Now, A, B, C, D are coplanar
`hArr[vec(AB)vec(AC)vec(AD)]=0hArr|(9,lambda-3,2),(11,4,1),(-7,14,-3)|=0`
`hArr 9 (-12-14)-(lambda-3)(-33+7)+2(154+28)=0`
`hArr - 234+26lambda - 78 + 364=0 hArr 26lambda=-52 hArr lambda =-2`.
Hence, the required value of `lambda ` is -2.