Well defined operations on vectors

Question

I am in the process of studying for a calc 3 exam and on all of the past exams there is a question like "Which of the following is a well-defined operation on vectors?" and I have no idea what this means, the only thing in the lecture notes I have been able to find was a footnote and google is turning up nothing for vectors. Thanks in advance

Answer 1

If $v, w, $ and $u$ are vectors, then $v\times w$ is a vector and $v\cdot w$ is a scalar.

Therefore, $a\times(b\times c)$ is well defined as the cross product of two vectors, $a$ and $b\times c$,

but $a\cdot(b\cdot c)$ is not well defined as the dot product of two vectors, because $b \cdot c$ is not a vector --

$b \cdot c$ is a scalar.