If `a=2hat(i)+2hat(j)+3hat(k),b=hat(i)+2hat(j)+hat(k) and c=3hat(i)+hat(j)` such that `a+lambda b` is perpendicular to c, then find the value of `lambda`.

Explanation

The given vectors are `a=2hat(i)+2hat(j)+3hat(k),b =-hat(i)+2hat(j)+hat(k) and c = 3hat(i)+hat(j)`
Now , `(a+lambda b) bot c`.
`rArr (a+lambda b).c=0`
(`because` Scalar product of two perpendicular vectors is zero)
`rArr [(2hat(i)+2hat(j)+3hat(k))+lambda(-1hat(i)+2hat(j)+hat(k))].(3hat(j)+hat(j))=0`.
`rArr [(2-lambda )hat(i)+(2+2lambda)hat(j)+(3+lambda)hat(k)].(3hat(i)+hat(j))=0`.
`rArr (2-lambda 3+(2+2lambda)1+(3+lambda )0=0`
`rArr 6-3lambda +2+2lambda =0rArr 8-lambda =0rArr lambda =8`.
Hence, the required value of `lambda` is 8.