Question

please prove this.it is a advanced mathematics question of class 10​

Answer 1

Explanation:

Let α⃗  and β⃗  be two unit vectors, and A and B be the angles made by them respectively with the X-axis.

So, α⃗ =cosAiˆ+sinAjˆβ⃗ =cosBiˆ+sinBjˆ

Now, α⃗ .β⃗ =(cosAiˆ+sinAjˆ)(cosBiˆ+sinBjˆ)

⇒αβcos(A−B)=cosAcosB+sinAsinB⇒cos(A−B)=cosAcosB+sinAsinB    [∵α=1, β=1] ------------(1)

Putting -B in place of B in (1):-

cos(A−(−B))=cosAcos(−B)+sinAsin(−B)⇒cos(A+B)=cosAcosB−sinAsinB

Similarly, 

α⃗ ×β⃗ =(cosAiˆ+sinAjˆ)×(cos