Priority of vector operators What is the precedence order for cross product, dot product and scalar multiplication?
$$x \overrightarrow {A}\cdot \overrightarrow {B} \times \overrightarrow {C} =\quad?$$
 A: 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.
A: Scalar multiplication can come at any time. If you would apply the dot product before the cross product, then the cross product would not be defined, as the dot product outputs a scalar, while the cross product needs two vectors.
Thus $$x\vec A \cdot \vec B \times \vec C = (x\vec A)\cdot(\vec B\times \vec C) = x(\vec A\cdot(\vec B\times \vec C)) $$
