First slide

Algebra of vectors

Question

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

Moderate

A

$\frac{\vec{a} - \vec{b}}{2 \cos (θ / 2)}$
B

$\frac{\vec{a} + \vec{b}}{2 \cos (θ / 2)}$
C

$\frac{\vec{a} - \vec{b}}{\cos (θ / 2)}$
D

none of these

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 $\frac{\vec{a} + \vec{b}}{| \vec{a} + \vec{b} |}$
From the figure, position vector of E is $\frac{\vec{a} + \vec{b}}{2}$
$Now in triangle AEB, AE = ABcos \frac{θ}{2}$
$\Rightarrow |\frac{\vec{a} + \vec{b}}{2}| = \cos \frac{θ}{2}$
Hence, unit vector along the bisector is $\frac{\vec{a} + \vec{b}}{2 \cos (θ / 2)}$

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