How would I answer the following vectors questions?

Question

1. is it possible to have a↓b undefined?
2. is it possible to have a↓b = b↓a?
3. explain why a↓c = a↓(b↓c)?

* all of a,b and c should have the arrow above them but I couldn’t find a way to add it *

Answer 1

I'm not certain whether you are talking about the cross product or dot product of vectors, so I'll give the answers as it pertains to both.

1. No, for both cross and dot products. The answers may be zero, but they will always be defined assuming that both a and b are legitimate vectors.
2. This will always be true for the dot product, but rarely true for the cross product. axb = - bxa meaning they would only be equal if the result is the zero vector. This would imply that a and b are parallel.
3. This is not true in general. The right hand side would be undefined if we're talking about the dot product as the dot product yields a scalar which cannot again be the input to a vector product.

If we're talking the cross product, the only way this could happen is if bxc were to lie in the plane defined by the vectors 'a' and 'c'.