The value of p so that vectors $\left(2\hat{i}-\hat{j}+\hat{k}\right),\left(\hat{i}+2\hat{j}-3\hat{k}\right)\text{and}\left(3\hat{i}+\text{p}\hat{j}+5\hat{k}\right)$ are coplanar should be :

$\left(\text{a}\times \text{b}\right).\text{c}=0$

$\text{a}\times \text{b}=\left|\begin{array}{ccc}i& j& k\\ 2& -1& 1\\ 1& 2& -3\end{array}\right|$

$=i\left(3-2\right)-j\left(-6-1\right)+k\left(4+1\right)$

$=i+7j+5k$

$\left(\text{a}\times \text{b}\right).\text{c}=0$

$=\left(i+7j+5k\right)\left(3i+\text{P}j+5k\right)$

$3+7P+25=0$

$7\text{P}+28=0$

$\text{P}=-4$