Precedence of cross product and dot product

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.

Solution 1:

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.