Question

If two unit vector A and B then prove that
|A|×|B|=sin a/2

Answer 1

Since a and b are unit vectors, |a| = 1, |b| = 1

Lets assume angle between the unit vectors, a and b, is x.

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

|a - b| = sqrt( |a|^2 + |b|^2 - 2 cosx)

=> |a - b| = sqrt( 1 + 1 - 2 cosx)

Since, cosx = 1 - 2 sin^2 (x/2)

=> |a - b| = sqrt( 2 - 2 + 4 sin^2 (x/2))

=> |a - b| = 2 sin(x/2)

=> sin(x/2) = 1/2 |a - b| Hence proved