Quick Math Cross Product Matrix

Cross Product Matrix

In vector algebra, the cross product is an operation on two vectors in three-dimensional space, producing another vector that is perpendicular to the plane containing the original vectors. This can be represented using a special matrix called the cross product matrix or skew-symmetric matrix.

Definition

Given a vector \(\mathbf{a} = [a_1, a_2, a_3]^T\), the cross product matrix \([ \mathbf{a} ]_\times\) is defined as:

\[
[\mathbf{a}]_\times = \begin{pmatrix}
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0
\end{pmatrix}
\]

Cross Product Using the Matrix

For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the cross product \(\mathbf{a} \times \mathbf{b}\) can be computed using the cross product matrix as follows:

\[
\mathbf{a} \times \mathbf{b} = [\mathbf{a}]_\times \mathbf{b}
\]

Example

Let’s compute the cross product of \(\mathbf{a} = [1, 2, 3]^T\) and \(\mathbf{b} = [4, 5, 6]^T\).

1. Construct the Cross Product Matrix:

\[
[\mathbf{a}]_\times = \begin{pmatrix}
0 & -3 & 2 \\
3 & 0 & -1 \\
-2 & 1 & 0
\end{pmatrix}
\]

2. Compute the Cross Product:

\[
\mathbf{a} \times \mathbf{b} = [\mathbf{a}]_\times \mathbf{b} = \begin{pmatrix}
0 & -3 & 2 \\
3 & 0 & -1 \\
-2 & 1 & 0
\end{pmatrix}
\begin{pmatrix}
4 \\
5 \\
6
\end{pmatrix}
\]

Perform the matrix-vector multiplication:

\[
= \begin{pmatrix}
0 \cdot 4 + (-3) \cdot 5 + 2 \cdot 6 \\
3 \cdot 4 + 0 \cdot 5 + (-1) \cdot 6 \\
-2 \cdot 4 + 1 \cdot 5 + 0 \cdot 6
\end{pmatrix}
= \begin{pmatrix}
0 – 15 + 12 \\
12 + 0 – 6 \\
-8 + 5 + 0
\end{pmatrix}
= \begin{pmatrix}
-3 \\
6 \\
-3
\end{pmatrix}
\]

Verification

To verify, we can use the standard cross product formula:

\[
\mathbf{a} \times \mathbf{b} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 2 & 3 \\
4 & 5 & 6
\end{vmatrix} = \mathbf{i}(2 \cdot 6 – 3 \cdot 5) – \mathbf{j}(1 \cdot 6 – 3 \cdot 4) + \mathbf{k}(1 \cdot 5 – 2 \cdot 4)
= \mathbf{i}(-3) – \mathbf{j}(-6) + \mathbf{k}(-3)
= [-3, 6, -3]
\]

Thus, the cross product calculated using the cross product matrix is verified. This example illustrates how to compute the cross product of two vectors using the cross product matrix approach.