Dot product $u\cdot (v\cdot w)$

Question

Let $u$,$v$, and $w$ be vectors. Does the expression below make sense?

$u\cdot (v\cdot w)$

Answer in the book says No but I didn't get it. $v\cdot w$ is scalar and we multiply it with a $u$ vector. can someone explain this please?

Answer 1

The issue is that you need to understand the meaning of the first dot in

$$u \cdot (v \cdot w).$$ If it means again the scalar product then indeed, the expression makes no sense.

If the first dot means the external multiplication of a scalar by a vector, then it could make sense. I say could as usually the scalar is first in the product, this is not the case in $u \cdot (v \cdot w).$

Answer 2

If the book says no, the first dot is to be taken as the dot product operator, and the expression is indeed meaningless.

Scalar multiplication is usually denoted without an operator, and preferably with the scalar on the left. $(v\cdot w)\,u$ would be ok, $u\,(v\cdot w)$ less common.

Also do not confuse with the mixed product,

$$u\cdot(v\times w).$$

Answer 3

Sort of bad notation. There is some reason to denote the inner product of two vectors as $\langle u,v\rangle$ instead of $u\cdot v$

Assuming the vectors are of the same dimension, $u\langle v,w\rangle$ is perfectly valid wheras $\langle u,\langle v,w\rangle\rangle$ is only defined if the the dimension of the vector is $1$.