Given three arbitrary vectors $\vec x_1, \vec x_2,\vec x_3$ in a (three dimensional) vector space, consider:
$\vec x_1\cdot \vec x_2 |\vec x_3|+\vec x_2\cdot \vec x_3 |\vec x_1|+\vec x_3\cdot \vec x_1 |\vec x_2|$
where $\cdot$ denotes the scalar product, and $|\vec x|$ denotes an absolute value of a vector.
Seeing that this combination looks kind of symmetric, does this have any geometrical meaning/interpretation? Is there some way to write it equivalently in terms of some other operations on $\vec x_1, \vec x_2,\vec x_3$?
It's $$|\vec{x_1}||\vec{x_2}||\vec{x_3}|\left(\cos\measuredangle\left(\vec{x_1},\vec{x_2}\right)+\cos\measuredangle\left(\vec{x_1},\vec{x_3}\right)+\cos\measuredangle\left(\vec{x_2},\vec{x_3}\right)\right).$$