Question

If $\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k}$ and $\hat{i}-6\hat{j}-\hat{k}$ are the position vectors of points $A,B,C$ and $D$ respectively, then find the angle between $\overrightarrow{AB}$ and $\overrightarrow{CD}$

Answer 1

$\overrightarrow{AB}=i+4j-k$ and $\overrightarrow{CD}=-2i-8j+2k$
Hence the angle between the vectors is given as
$\theta =co{s}^{-1}\left(\frac{\overrightarrow{AB}.\overrightarrow{CD}}{AB.CD}\right)$

$=co{s}^{-1}\left(\frac{-2-32-2}{\sqrt{18}\times \sqrt{72}}\right)$

$=co{s}^{-1}\left(\frac{-36}{36}\right)$

$=co{s}^{-1}(-1)$
$=\pi$