If $3\hat{\mathrm{i}}-5\hat{\mathrm{j}}+2\hat{\mathrm{k}}, 7\hat{\mathrm{i}}+2\hat{\mathrm{j}}-4\hat{\mathrm{k}}, \hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ and $-7\hat{\mathrm{i}}-17\hat{\mathrm{j}}+16\hat{\mathrm{k}}$ are position vectors of the points $A\mathit{,}B,C$ and $D$ respectively, then the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is

Given,

$3\hat{\mathrm{i}}-5\hat{\mathrm{j}}+2\hat{\mathrm{k}}, 7\hat{\mathrm{i}}+2\hat{\mathrm{j}}-4\hat{\mathrm{k}}, \hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ and $-7\hat{\mathrm{i}}-17\hat{\mathrm{j}}+16\hat{\mathrm{k}}$ are position vectors of the points $A\mathit{,}B,C$ and $D$ respectively,

Then $\overrightarrow{AB}=\left(7-3\right)\hat{\mathrm{i}}+\left(2+5\right)\hat{\mathrm{j}}-\left(4+2\right)\hat{\mathrm{k}}$

$\Rightarrow \overrightarrow{AB}=4\hat{\mathrm{i}}+7\hat{\mathrm{j}}-6\hat{\mathrm{k}} .......\left(1\right)$

And $\overrightarrow{CD}=\left(-7-1\right)\hat{\mathrm{i}}+\left(-17+3\right)\hat{\mathrm{j}}+\left(16-4\right)\hat{\mathrm{k}}$

$\Rightarrow \overrightarrow{CD}=-8\hat{\mathrm{i}}+-14\hat{\mathrm{j}}+12\hat{\mathrm{k}} ..........\left(2\right)$

Then the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is given by,

$\cos \theta =\left|\frac{\overrightarrow{AB}.\overrightarrow{CD}}{\left|\overrightarrow{AB}\right|\left|\overrightarrow{CD}\right|}\right|$

Also we can see that $\overrightarrow{CD}=-2||\overrightarrow{AB}||$ from equation $\left(1\right) \& \left(2\right)$

So angle between them will be $\mathrm{\pi }$.