Multivariable calculus and vector analysis

Scalar triple product example

Example

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

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

The triple product is

$$\begin{array}{llll}\hfill \left(\mathbf{a}\times \mathbf{b}\right)\cdot \mathbf{c}& =\left|\begin{array}{ccc}\hfill -1& \hfill 3& \hfill 3\\ \hfill -2& \hfill 3& \hfill 1\\ \hfill 0& \hfill 4& \hfill 0\end{array}\right|& \hfill & \\ \hfill & =-1\left(0-4\right)-3\left(0-0\right)+3\left(-8+0\right)& \hfill & \\ \hfill & =4-24=-20& \hfill & \end{array}$$

Hence the volume is $|-20|=20$.

Below is the CVT used to illustrate the scalar triple product. However, here the vectors fixed at the above values.

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

• The cross product
• The component formula for the cross product
• The scalar triple product
• Cross product examples
• Scalar triple product example
• Spherical coordinates
• Examples of n-dimensional vectors
• Function notation

1b. Lines and planes

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material