Find angle between the vectors a=i+j-k and b=i-j+k - Vidyadip

Question

Find angle $\theta$ between the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$

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Answer

The angle $\theta$ between two vectors $\vec a$ and $\vec b$ is given by
$\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
Now $\vec{a} \cdot \vec{b}=(\hat{i}+\hat{j}-\hat{k}) \cdot(\hat{i}-\hat{j}+\hat{k})=1-1-1=-1$
Therefore, we have $\cos \theta=\frac{-1}{3}$
hence the required angle is $\theta=\cos ^{-1}\left(-\frac{1}{3}\right)$

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