### Scalar triple product example

#### Example

Find the volume of the parallelepiped spanned by the vectors $\vc{a} = (-2,3,1)$, $\vc{b} = (0, 4, 0)$, and $\vc{c} = (-1,3,3)$.

**Solution:** The volume is the absolute value of the
scalar triple product of the three vectors.

The triple product is \begin{align*} (\vc{a} \times \vc{b}) \cdot \vc{c} & =\left| \begin{array}{rrr} -1 & 3 & 3\\ -2 & 3 & 1\\ 0 & 4 & 0 \end{array} \right| \\ &= -1(0-4)- 3 (0-0) +3(-8+0) \\ &= 4-24 = -20 \end{align*} Hence the volume is $|-20| = 20$.

Below is a modified version of the applet used to illustrate the scalar triple product. In this case, the vectors have been fixed to be the values of this example.

#### Applet loading

*Scalar triple product with fixed values.* The cyan slider shows the value of the product $(\color{blue}{\vc{a}} \times \color{green}{\vc{b}}) \cdot \color{magenta}{\vc{c}}$, where the vectors $\color{blue}{\vc{a}}$ (in blue), $\color{green}{\vc{b}}$ (in green), and $\color{magenta}{\vc{c}}$ (in magenta) are fixed to given values. The volume of the spanned parallelepiped (outlined) is the magnitude $\|(\color{blue}{\vc{a}} \times \color{green}{\vc{b}}) \cdot \color{magenta}{\vc{c}}\|$. The cross product $\color{blue}{\vc{a}} \times \color{green}{\vc{b}}$ is shown by the red vector. The three-dimensional perspective of this graph is hard to perceive when the graph is still. If you keep the figure rotating by dragging it with the mouse, you'll see it much better.

#### Thread navigation

##### Vector algebra

- Previous: The scalar triple product
- Next: Introduction to matrices

##### Math 2374

#### Similar pages

- The scalar triple product
- The cross product
- Cross product examples
- The formula for the cross product
- The dot product
- The formula for the dot product in terms of vector components
- Dot product examples
- The relationship between determinants and area or volume
- Multiplying matrices and vectors
- Matrix and vector multiplication examples
- More similar pages