Question

If the vector $-3\hat{i}+4\hat{j}-2\hat{k},\hat{i}+2\hat{k},\hat{i}-p\hat{j}$ are coplanar, then the value of $p$ is

• A. $-2$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

Answer: (D)

$2$

The given vectors are $-3\hat{i}+4\hat{j}-2\hat{k},\hat{i}+2\hat{k}$ and $\hat{i}-p\hat{j}$.
Let $\overrightarrow{a}=-3\hat{i}+4\hat{j}-2\hat{k}$
$\overrightarrow{b}=\hat{i}+2\hat{k}$
and $\overrightarrow{c}=\hat{i}-p\hat{j}$.
If the vector $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are coplanar, then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=0$
$\Rightarrow \left|\begin{array}{ccc}-3& 4& -2\\ 1& 0& 2\\ 1& -p& 0\end{array}\right|=0$
$\Rightarrow -3(0+2p)+4(2-0)-2(-p-0)=0$
$\Rightarrow -6p+8+2p=0$
$\Rightarrow -4p+8=0$
$\Rightarrow 4p=8$
$\Rightarrow p=2$
Hence, the correct answer from the given alternatives is option D.