Question

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

• A. $\frac{\vec{a}-\vec{b}}{2\cos (\theta /2)}$
• B. $\frac{\vec{a}+\vec{b}}{2\cos (\theta /2)}$
• C. $\frac{\vec{a}-\vec{b}}{\cos (\theta /2)}$
• D. none of these

Answer 1

Answer: (B)

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 $\frac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}$.

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 $\frac{\vec{a}+\vec{b}}{2\cos (\theta /2)}$.