The scalar triple product of three vectors a, b, and c, is a·(b×c), the scalar product of a with the vector product b×c. It is a scalar quantity and is denoted [a,b,c]. It has the following properties:

1. (i) [a,b,c] = −[a,c,b].

2. (ii) [a,b,c] = [b,c,a] = [c,a,b].

3. (iii) The vectors a, b, and c are linearly dependent if and only if [a,b,c] = 0.

4. (iv) If a = (a₁,a₂,a₃), etc., then

   $$\begin{array}{l}[a,b,c]=a_1(b_2{c}_3-b_3{c}_2)\hfill \\ +a_2(b_3{c}_1-b_1{c}_3)+a_3(b_1{c}_2-b_2{c}_1)\hfill \\ =\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right].\hfill \end{array}$$

5. (v) Let $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ represent a, b and c. Then the parallelepiped with OA, OB, and OC as three of its edges has volume equal to the absolute value of [a,b,c].