Properties of Vector Product Algebra?

Question

Is the following vector cross product identity the Associative Law or the Distributive Law of scalar multiplication or neither

p a x q b = p q (a x b),

where p and q are scalars and a and b are non-zero vectors.

Please help, I am really confused!

Answer 1

The distributive law would look like a x (b + c) = a x b + a x c

or a.(b x c) = a.b x a.c



The associative law would be (a x b) x c = a x (b x c) which doesn't exist in vector product . Even a.(b x c) = (a.b) x c isn't possible as one is a number and the other is a vector



The second example of the distributive law seems very close to your question

Answer 2

Just write out the vectors: u = (u1, u2, ... , un) v = (v1, v2, ... , vn) Then u (dot) v = u1v1 + u2v2 + ... + unvn = v1u1 + v2u2 + ... + vnun = v (dot) u. It's follows from the fact that scalar multiplacation is commutative (that is, 3 * 4 = 4 * 3). You can prove 2) in the same way.

Answer 3

Neither. Why bother with what law it is----just carry out the operation?

p a x q b = pq (a x b) = qp (a x b) if required

Note

pq = qp (scalars)

(a x b) = - (b x a) (vectors)

Answer 4

Neither

Answer 5

Associative Law

if it were distributive it will be

p a x q b = pqa x pqb,

Answer 6

distributive