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Problem :

Find the vector $\vec{c}$ directed along the bisector of the angle between the vectors $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ where $|\vec{c}|=\sqrt{2}$

Please suggest how to proceed in such problems , no idea how to find bisector of angle, however done in straight lines but not in vectors. please suggest will be of great help. thanks.

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Comsider the unit vectors $$\hat{a}=\frac{1}{\sqrt{6}}\left(\begin{matrix}1\\2\\1\end{matrix}\right)$$ And $$\hat{b}=\frac{1}{\sqrt{6}}\left(\begin{matrix}2\\-1\\1\end{matrix}\right)$$

Then the vector in the direction either internally or externally bisecting the angle between $a$ and $b$ is parallel to $$\hat{a}\pm\hat{b}$$

So you can calculate either of these, divide by the modulus and multiply by $\sqrt{2}$ and you have what you want.

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