The scalar triple product

The scalar triple product of three vectors a, b, and c is (a × b) · c. It is a scalar product because it evaluates to a single number (unlike the cross product). The scalar triple product is important because its absolute value |(a × b) · c| is the volume of the parallelepiped spanned by a, b, and c (i.e., the parallelepiped whose adjacent sides are the vectors a, b, and c).

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This results from simple geometry. The volume is the area of the base times the height. We already know the area of the parallelogram base: ||a × b||. The height is the component of c in the direction normal to the base, i.e., in the direction of a × b. Hence the height is $\left\vert\vphantom{ \vert\vert\mathbf{c}\vert\vert \cos \phi }\right.$||c|| cos$\phi$$\left.\vphantom{ \vert\vert\mathbf{c}\vert\vert \cos \phi }\right\vert$, where $\phi$ is the angle between c and a × b. (Why do we need the absolute value? cos$\phi$ would be negative if $\phi$ > $\pi$/2.)

The volume of the parallelepiped is therefore

$$\left\vert\vphantom{ \vert\vert\mathbf{a} \&\char93 215; \mathbf{b}\vert\vert \vert\vert\mathbf{c}\vert\vert \cos \phi}\right. \phi \left.\vphantom{ \vert\vert\mathbf{a} \&\char93 215; \mathbf{b}\vert\vert \vert\vert\mathbf{c}\vert\vert \cos \phi}\right\vert$$

(Remember the definition of the dot product.) Using the formula for the cross product in components form, we can write the scalar triple product in component form as Write scalar triple product in component form as

$$\left\vert\vphantom{ \begin{array}{cc} a_2 & a_3 b_2 & b_3 \end{array} }\right. \begin{array}{cc} a_2 & a_3 b_2 & b_3 \end{array} \left.\vphantom{ \begin{array}{cc} a_2 & a_3 b_2 & b_3 \end{array} }\right\vert \left\vert\vphantom{ \begin{array}{cc} a_1 & a_3 b_1 & b_3 \end{array} }\right. \begin{array}{cc} a_1 & a_3 b_1 & b_3 \end{array} \left.\vphantom{ \begin{array}{cc} a_1 & a_3 b_1 & b_3 \end{array} }\right\vert \left\vert\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right. \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} \left.\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right\vert$$ (a × b) · c

$$\left\vert\vphantom{ \begin{array}{ccc} c_1 & c_2 & c_3 a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array} }\right. \begin{array}{ccc} c_1 & c_2 & c_3 a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array} \left.\vphantom{ \begin{array}{ccc} c_1 & c_2 & c_3 a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array} }\right\vert$$

Why did I repeat this proof in these notes? To help you understand what is going on when you view this Concept-Visualization Tool (CVT) of the triple scalar product.

In this CVT, the vector a is blue, b is green, c is magenta (shown with balls on their ends). The CVT includes the outline of the parallelepiped spanned by these vectors, whose volume is |(a × b) · c|.

For illustration, I've drawn the cross product a  ×  b in red. Except for sign, the volume of the parallelepiped is the dot product between c and the cross product. (I must apologize to color blind folks for such reliance on colors. I hope you can figure out what is going on even if the correspondence betweent the colored vectors and a, b, c isn't clear.)

The value of the scalar triple product (a × b) · c is shown by the cyan slider (which you cannot drag). Note that the scalar triple product can be positive, negative, or zero. (That's why we need the absolute value for the volume.) What determines the sign of (a × b) · c? Also, when (a × b) · c = 0, what is going on? (If you rotate the graph once you've made triple scalar product zero, you'll immediately see that answer.)

Important note: The three-dimensional perspective of this graph is hard to perceive when the graph is still. If you keep the figure rotating, you'll see it much better.

Click and drag the blue, green, and magenta balls to change the vectors.

In case you like to see it with numbers, here's an example of calculating the volume of a parallelepiped using the triple product.