Answer
0
Work Step by Step
$(\vec{d_{1}}+\vec{d_{2}})=(3+(-5))\hat{i}+(-2+2)\hat{j}+(4-1)\hat{k}$
$=-2\hat{i}+0\hat{j}+3\hat{k}=-2\hat{i}+3\hat{k}$
$4\vec{d_{2}}=(4\times-5)\hat{i}+(4\times2)\hat{j}+(4\times-1)\hat{k}$
$=-20\hat{i}+8\hat{j}-4\hat{k}$
$(\vec{d_{1}}\times4\vec{d_{2}})=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\3&-2&4\\-20&8&-4\end{vmatrix}=-24\hat{i}-68\hat{j}-16\hat{k}$
Now, $(\vec{d_{1}}+\vec{d_{2}})\cdot(\vec{d_{1}}\times4\vec{d_{2}})=(-2\hat{i}+3\hat{k})\cdot(-24\hat{i}-68\hat{j}-16\hat{k})$
$=(-2\times-24)+(0\times-68)+(3\times-16)=48+0-48=0$
