TITLE: Precedence of cross product and dot product
QUESTION [10 upvotes]: Which operator precedence is higher? The one of the cross product or the one of the dot product?
Consider the following term:
$$\overrightarrow {A},
\overrightarrow {B},
\overrightarrow {C},
\overrightarrow {D} \in \mathbb{R}^n$$
$$\overrightarrow {A}\times \overrightarrow {B} \cdot \overrightarrow {C} \times \overrightarrow {D} = \overrightarrow {X}$$
Should this be
$$\overrightarrow {A}\times (\overrightarrow {B} \cdot \overrightarrow {C}) \times \overrightarrow {D} = \overrightarrow {X}$$
$$\overrightarrow {X} \in \mathbb{R}^n$$
or
$$(\overrightarrow {A}\times \overrightarrow {B}) \cdot (\overrightarrow {C} \times \overrightarrow {D}) = \overrightarrow {X}$$
$$\overrightarrow {X} \in \mathbb{R}$$
Both terms are valid but have an ambiguous result.

REPLY [8 votes]: The former is not valid. $B \cdot C$ yields a scalar, and you can't do cross product between a scalar and a vector.
Generally $\times$ has higher precedence because otherwise many operations don't make sense.