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

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Q. If $\vec{a}$ and $\vec{a}$ 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 $\frac{\vec{a} + \vec{b}}{\left|\vec{a}+\vec{b}\right|}$
image
From the figure, position vector of E is $\frac{\vec{a} + \vec{b}}{2} $
Now in triangle $AEB, AE = AB$ 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 \left(\theta/2\right)}$