Question

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

• A. $-2$
• B. $2$
• C. $0$
• D. $-1$

Answer 1

The correct option is D $-1$

Since the given vectors are coplaner, their triple product is 0.

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

$\Rightarrow p(qr-1)-1(r-1)+1(1-q)=0$

$\Rightarrow pqr-p-r+1-q=0$

$\Rightarrow pqr-(p+q+r)=-1$