TITLE: Priority of vector operators
QUESTION [5 upvotes]: What is the precedence order for cross product, dot product and scalar multiplication?
$$x \overrightarrow {A}\cdot \overrightarrow {B} \times \overrightarrow {C} =\quad?$$

REPLY [7 votes]: First note that the cross product is an operation between two vector that gives a vector as result and the dot product is an operation between two vectors that gives a scalar as result. So the mixed product
$$
A\cdot B \times C
$$
has a sense only in the oreder $A\cdot(B\times C)$ and has a scalar as a result.
For the scalar product by a real number $x$ we know that both the dot and cross product are compatible with scalar multiplication, this means that:
$$
x(A\cdot B)= (xA)\cdot B = A \cdot (xB)
$$
and
$$
x(A\times B)= (xA)\times B = A \times (xB)
$$
so in the expression $ xA\cdot B\times C$ we can perform the scalar multiplication when we want ( but only one time for one vector) but we have to calculate the cross product before the dot product.