Vector Algebra Question 206
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
Options:
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
Show Answer
Answer:
Correct Answer: B
Solution:
Vector in the direction of angular bisector of $ \vec{a} $ and $ \vec{b} $ is $ \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} $ . Unit vector in this direction is $ \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} $ . From the figure, position vector of E is $ \frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|} $ Now in triangle $ AEB,AE=AB\cos \frac{\theta }{2} $ $ \Rightarrow | \frac{\vec{a}+\vec{b}}{2} |=\cos \frac{\theta }{2} $ Hence, unit vector along the bisector is $ \frac{\vec{a}+\vec{b}}{2|\cos (\theta /2)|} $ .
BETA
