Question

Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.

Answer 1

Let `hat"OP"` and `hat"OQ"` be unit vectors making angles A and B, respectively, with positive direction of x-axis.

Then ∠QOP = A – B  .....[From the figure]

We know `hat"OP" = vec"OM" + vec"MP"`

= `hat"i" cos "A" + hat"j" sin "A"` and `hat"OQ" = vec"ON" + vec"NQ" = hat"i" cos"B" + hat"j" cos"B"`.

 By definition `hat"OP" * hat"QO" = |hat"OP"| |hat"OQ"| cos("A" - "B")`

= `cos("A" - "B")`   ......(1) `(because hat"OP"|= 1 = |hat"OQ"|)`

In terms of components, we have

`hat"QP" * hat"OQ" = (hat"i" cos"A" + hat"j" sin "A")*(hat"i" cos"B" + hat"j" sin"B")`

= cosA cosB + sinA sinB  .....(2)

From (1) and (2), we get

cos(A – B) = cosA cosB + sinA sinB.