Question

Prove, by the vector method, sin (A – B) = sinA cosB – cosA sinB.

Answer 1

Here, vector OP = ai + bj

and vector OQ = ci + dj

Now, OP × OQ = (ad – bc)k

[(OP)(OQ)sin(∠POQ)] = (ad – bc)k

–sin(A – B) = (ad – bc)/(OP)(OQ)

= (a/OP).(d/OQ).(b/OP).(c/OQ)

= cosA.sinB – sinA.cosB

sin(A – B) = sinA.cosB – cosA.sinB

Hence, the result.