Question

Choose the correct answer
Let a and b be two unit vectors and $\theta$ is the angle between them. Then, a + b is a unit vector, if

a) $\theta =\frac{\pi }{4}$
b) $\theta =\frac{\pi }{3}$
c) $\theta =\frac{\pi }{2}$
d) $\theta =\frac{2\pi }{3}$

Answer 1

Let a and b be two unit vectors and $\theta$ be the angle between them.
Then, $|a|=|b|=1$
Now, (a + b) is a unit vector if
$|a+b|=1\Rightarrow (a+b{)}^2=1\Rightarrow (a+b).(a+b)=1\Rightarrow a.a+a.b+b.a+b.b=1$
$\Rightarrow |a{|}^2+|b{|}^2+2a.b=1$ $[\because a.b=b.a]$
$\Rightarrow 1^2+1^2+2a.b=1\Rightarrow a.b=-\frac{1}{2}\Rightarrow |a||b|cos\theta =-\frac{1}{2}$
$\Rightarrow 1\times 1\times cos\theta =-\frac{1}{2}\Rightarrow cos\theta =-\frac{1}{2}\Rightarrow \theta =\frac{2\pi }{3}$