Triple Product

a \cdot (b \times c) = det a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}

a \cdot (b \times c) = ∥ a ∥∥ b ∥∥ c ∥

Applications

$a \cdot (b \times c)$ is the volume of a parallelepiped where the vectors are the three lengths (in any order)
Coplanar Vectors: $a, b, c \in P ⟺ a \cdot (b \times c) = 0$
- Because the parallelepiped they form has no volume when they are all on the same plane
- Can also be used for points (connect them with vectors): $A, B, C, D \in P ⟺ A B \cdot (A C \times A D) = 0$
- Also means that $a \cdot (b \times c) = 0 ⟺ c = s a + t b$ because of how coplanar vectors work

Cyclic permutations: $a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$
Other permutations: $a \cdot (b \times c) = - b \cdot (a \times c) = \dots$