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  4. If veca and vecb are two unit vectors and θ is the angle between them, then the unit vector along the angular bisector of veca and vecb will be given by

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Q. If $\vec{a}$ and $\vec{b}$ are two unit vectors and $\theta$ is the angle between them, then the unit vector along the angular bisector of $\vec{a}$ and $\vec{b}$ will be given by

Vector Algebra

Solution:

Vector in the direction of angular bisector of $\vec{a}$ and $\vec{b}$ is $\frac{\vec{a}+\vec{b}}{2}$
Unit vector in this direction is $\frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}$.
image
From the figure, position vector of $E$ is $\frac{\vec{a}+\vec{b}}{2}$
Now in triangle $A E B, A E=A B \cos \frac{\theta}{2}$
$\Rightarrow\left|\frac{\vec{a}+\vec{b}}{2}\right|=\cos \frac{\theta}{2}$
Hence, unit vector along the bisector is $\frac{\vec{a}+\vec{b}}{2 \cos (\theta / 2)}$.