Question

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

• A. $\frac{\vec{a}-\stackrel{⃗}{}wb}{2cos\left(\theta /2\right)}$
• B. $\frac{\vec{a}+\vec{b}}{2cos\left(\theta /2\right)}$
• C. $\frac{\vec{a}-\vec{b}}{cos\left(\theta /2\right)}$
• 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 $\frac{\vec{a}+\vec{b}}{2cos\left(\theta /2\right)}$