Question

If the vector $p\hat{i}+\hat{j}+\hat{k},\hat{i}+q\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r\hat{k}\left(p\ne q\ne r\ne 1\right)$ are coplanar, then the value of $pqr-\left(p+q+r\right)$ is

• A. $2$
• B. $0$
• C. $-1$
• D. $-2$

Answer 1

Answer: (A)

$2$

If vector $p\hat{i}+\hat{j}+\hat{k},\hat{i}+q\hat{i}+\hat{k},\hat{i}+\hat{j}+r\hat{k}$ are

Coplanar then

$\left|\begin{array}{ccc}p& 1& 1\\ 1& q& 1\\ 1& 1& r\end{array}\right|=0$

$pqr-p-1(r-1)+1(1-q)=0$

$pqr-p-q-r=-2$

$pqr-(p+q+r)=-2$

$D$ is correct.