If the vectors $p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$…
If the vectors $p \hat{i}+\hat{j}+\hat{k}, \hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$ $(\mathrm{p} \neq \mathrm{q} \neq \mathrm{r} \neq 1)$ are coplanar, then the value of $\mathrm{pq} \mathrm{r}-(\mathrm{p}+\mathrm{q}+\mathrm{r})$ is
$-2$
2
0
$-1$
Solution
Since $\mathrm{pi}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\mathrm{q} \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{rk}$ are coplanar,
$\left|\begin{array}{lll}p & 1 & 1 \\ 1 & q & 1 \\ 1 & 1 & r\end{array}\right|=0$
$\begin{aligned} & \Rightarrow p(q r-1)-1(r-1)+1(1-q)=0 \\ & \Rightarrow p q r-p-q-r+2=0 \\ & \Rightarrow p q r-(p+q+r)=-2\end{aligned}$