Given the vectors u, v and w in R3, which of the following calculations could result in the answer 3?

Question

Given the vectors u, v and w in R3, which of the following calculations could result in the answer 3?

1. (u x v) • (v x w)
2. u x (u - w)
3. (u • v) (u + w)
4. ||u + v|| w

Answer 1

The number is 3 isn't important, what is important is that 3 is a scalar quantity. So we are seeking an expression involving vectors that reduces to a scalar.

1. (u x v) • (v x w) = vector * vector = scalar
2.  u x (u - w) = vector x vector = vector
3. (u • v) (u + w) = scalar * vector = vector
4. ||u + v|| w = scalar * vector = vector

So only the first produces a scalar, so that's the answer.