1.- 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

I am assuming that we are saying that 1, 2, and 3 are the magnitude of a, b, and c. Then, for them to sum to 0, the vectors of length 1 and 2 in the opposite direction to the vector of length 3. Then, |a+b+c|^2 = a.a + b.b + c.c + 2(a.b+a.c+b.c), or

a.b+a.c+b.c = ((a+b+c)^2-a.a -b.b -c.c)/2 =

(0^2-1^2-2^2-3^2)/2 =

-7