If the vectots `phati+hatj+hatk, hati+qhatj+hatk` and `hati+hatj+rhatk(p!=q!=r!=1)` are coplanar, then the value of `pqr-(p+q+r)`, is

Options

A. `-2`
B. 2
C. 0
D. `-1`

Answer

A

Explanation

Let `a = phati + hatj + hatk, b = hati + hatj + hatk` and c = `hati + hatj + rhatk` are coplanar and `( p ne q ne r ne 1)`.
Since, a, b and c are coplanar .
`:. [a,b,c]= 0 rArr |{:(p,1,1),(1,q,1),(1,1,r):}| = 0`
`rArr p(qr- 1) - 1 (r-1) + 1 (1-q) = 0`
`rArr pqr - p -r + 1 + 1 -q = 0`
`:. pqr - (p+q+r) = - 2`