Triple Scalar Product

The cross product between two vectors dotted with another vector happens so frequently in linear algebra that it has its own special name: scalar triple product. To truly realize why this expression is so common and important I recommend looking at the visual illustration I drew in ms paint below. The triple scalar product provides the signed volume of the parallelepiped in 3d space. It's important to note the signed part because often in linear algebra a signed area or volume can tell you information about the vectors and their orientation to one another (clockwise or counter-clockwise). In addition, it may also be able to cheaply check if a line is within a triangle by checking which side the line lies on compared to the vertices of a triangle (figure 2).

(u × v) • w = (v × w) • u

 Figure 1. Triple Scalar Product visually.

Figure 2. Comparing a line to check if it's within a triangle.