If and are two-unit vectors inclined at an angle θ, then prove that(i) (ii)

Question

If  and  are two-unit vectors inclined at an angle θ, then prove that(i) (ii)

Philoid · Accepted Answer

Here, and are two-unit vectors then

Let us Assume, the angle between the units vector, a and b is θ

Now, Using the law of cosines on the triangle formed by vector a,b and its resultants

We know, cos x = 1-2 sin²(x/2)

Hence, Proved

(ii)Now,

And,

We know, 1+cos θ = 2 cos²θ/2

From part (i)

Dividing both we get

Hence, Proved

If and are two-unit vectors inclined at an angle θ, then prove that
(i)

(ii)