Let O be the origin and the position vector of the point P be - i ^ - 2 j ^ + 3 k . If the position vectors…

Given,

$O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3\hat{k}$,

So, $\overrightarrow{OP}=-\hat{i}-2\hat{j}+3\hat{k}$

Also given the position vectors of the points $A, B$ and $C$ are $-2\hat{i}+\hat{j}-3\hat{k}, 2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}^+2\hat{j}-\hat{k}$ 

Now finding, 

$\overrightarrow{AB}=4\hat{i}+3\hat{j}+\hat{k}$

$\overrightarrow{AC}=-2\hat{i}+\hat{j}+2\hat{k}$

And $\overrightarrow{AB}\times \overrightarrow{AC}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 4& 3& 1\\ -2& 1& 2\end{array}\right|$

$\Rightarrow \overrightarrow{AB}\times \overrightarrow{AC}=5\hat{i}-10\hat{j}+10\hat{k}$

Now finding the Projection of $\overrightarrow{OP}$ on vector perpendicular to $\overrightarrow{AB} \& \overrightarrow{AC}$ we get,

Projection$=\left|\frac{\overrightarrow{OP}·\left(\overrightarrow{AB}\times \overrightarrow{AC}\right)}{\left|\overrightarrow{AB}\times \overrightarrow{AC}\right|}\right|$

$=\frac{5\left(-\hat{i}-2\hat{j}+3\hat{k}\right)\left(\hat{i}-2\hat{j}+2\hat{k}\right)}{5\sqrt{1+4+4}}$$=3$