Discussion Forum

If a and b are unit vectors and $\alpha$  is the angle between  them, then a+b is a unit vector when $\cos \alpha$=$

Answer:

Explanation:

We have, |a|=|b|=1 and $\alpha$ is the angle between a and b

 Clearly , $\cos \alpha=\frac{a.b}{|a|.|b|}$

$\Rightarrow$  $\cos \alpha=a.b$   .........(i)

 Now, let a+b is a unit vector , then

       |a+b|=1

$\Rightarrow$     $|a+b|^{2}=1$

$\Rightarrow$   $(a+b).(a+b)=1$

$\Rightarrow$   $a.a+a.b+b.a+b.b=1$

$\Rightarrow$   $|a|^{2}-2a.b+|b|^{2}=1$     $[ \because a.b=b.a]$

$\Rightarrow$    $1+2\cos \alpha+1=1$   [using Eq.(i)]

$\Rightarrow$     $1+2\cos \alpha=0$

$\Rightarrow$  $\cos \alpha= -\frac{1}{2}$