The vectors a, b and c are such that a + b + c = 0. Determine the value of a·b + a·c + b·c if a=1, b=2, and c=3

Question

The vectors a, b and c are such that a + b + c = 0. Determine the value of a·b + a·c + b·c if ⎢a⎢=1, ⎢b⎢=2, and ⎢c⎢=3

Answer 1

b + c = - a

a•b + a•c = - a•a = - IaI² = - 1 (1)

a + b = - c

a•c + b•c = - c•c = - IcI² = - 9 (2)

a + c = - b

a•b +c•b = -b•b = - IbI² = - 4 (3)

By adding (1), (2) and (3) we have:

2(a•b + a•c + b•c) = - 14; from here a•b + a•c + b•c = - 7.