Multivariable calculus and vector analysis

Cross product examples

Example 1

Calculate the cross product between $\mathbf{a}=\left(3,-3,1\right)$ and $\mathbf{b}=\left(4,9,2\right)$.

Solution: The cross product is

$$\begin{array}{llll}\hfill \mathbf{a}\times \mathbf{b}& =\left|\begin{array}{ccc}\hfill \mathbf{i}\hfill & \hfill \mathbf{j}\hfill & \hfill \mathbf{k}\hfill \\ \hfill 3\hfill & \hfill -3\hfill & \hfill 1\hfill \\ \hfill 4\hfill & \hfill 9\hfill & \hfill 2\hfill \end{array}\right|& \hfill & \\ \hfill & =\mathbf{i}\left(-3\cdot 2-1\cdot 9\right)-\mathbf{j}\left(3\cdot 2-1\cdot 4\right)+\mathbf{k}\left(3\cdot 9+3\cdot 4\right)& \hfill & \\ \hfill & =-15\mathbf{i}-2\mathbf{j}+39\mathbf{k}& \hfill & \end{array}$$

Example 2

Calculate the area of the parallelogram spanned by the vectors $\mathbf{a}=\left(3,-3,1\right)$ and $\mathbf{b}=\left(4,9,2\right)$.

Solution: The area is $\parallel \mathbf{a}\times \mathbf{b}\parallel$. Using the above expression for the cross product, we find that the area is $\sqrt{{15}^2+2^2+{39}^2}=5\sqrt{70}$.

Example 3

Calculate the area of the parallelogram spanned by the vectors $\mathbf{a}=\left(3,-3,1\right)$ and $\mathbf{c}=\left(-12,12,-4\right)$.

Solution:

$$\begin{array}{llll}\hfill \mathbf{a}\times \mathbf{c}& =\left|\begin{array}{ccc}\hfill \mathbf{i}\hfill & \hfill \mathbf{j}\hfill & \hfill \mathbf{k}\hfill \\ \hfill 3\hfill & \hfill -3\hfill & \hfill 1\hfill \\ \hfill -12\hfill & \hfill 12\hfill & \hfill -4\hfill \end{array}\right|& \hfill & \\ \hfill & =\left(0,0,0\right)& \hfill & \end{array}$$

The magnitude of the zero vector is zero, so the area of the parallelogram is zero. What happened?

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