What Matrix Representation Is Used to Calculate the Cross Product?

The cross product of two three-dimensional vectors is calculated using the determinant of a specific 3x3 matrix. This special matrix representation provides a structured way to derive the resulting vector.

Understanding the Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. The result is a vector that is perpendicular to both of the input vectors and normal to the plane containing them. Its magnitude is equal to the area of the parallelogram spanned by the two vectors.

If you have two vectors, $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$, their cross product $\mathbf{a} \times \mathbf{b}$ is another vector.

Calculating the Cross Product Using a Matrix Determinant

As explained by the reference, the cross product can be expressed in the form of a determinant of a special 3 × 3 matrix. This matrix is constructed as follows:

The first row consists of the standard basis vectors: $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.
The second row consists of the scalar components of the first vector ($\mathbf{a}$): $a_1$, $a_2$, and $a_3$.
The third row consists of the scalar components of the second vector ($\mathbf{b}$): $b_1$, $b_2$, and $b_3$.

The matrix looks like this:

$$
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
a_1 & a_2 & a_3 \
b_1 & b_2 & b_3
\end{vmatrix}
$$

To find the cross product $\mathbf{a} \times \mathbf{b}$, you calculate the determinant of this matrix. This is often done using cofactor expansion along the first row or, for a 3x3 matrix, using Sarrus's rule. According to the reference, Sarrus's rule involves multiplications between matrix elements identified by crossed diagonals.

Expanding the determinant yields:

$\mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$

The result is a new vector whose components are the coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.

Sarrus's Rule Explained Briefly

Sarrus's rule is a mnemonic for calculating the determinant of a 3x3 matrix. You rewrite the first two columns of the matrix to the right of the original matrix. Then, you sum the products of the elements on the three main diagonals from upper-left to lower-right, and subtract the sum of the products of the elements on the three off-diagonals from upper-right to lower-left.

| i  j  k | i  j
| a1 a2 a3| a1 a2
| b1 b2 b3| b1 b2

Determinant = ($\mathbf{i}a_2b_3 + \mathbf{j}a_3b_1 + \mathbf{k}a_1b_2$) - ($\mathbf{k}a_2b_1 + \mathbf{i}a_3b_2 + \mathbf{j}a_1b_3$)

This simplifies to the cross product formula derived above.

Example Calculation

Let's find the cross product of $\mathbf{a} = 2\mathbf{i} + 1\mathbf{j} - 3\mathbf{k}$ and $\mathbf{b} = 4\mathbf{i} - 2\mathbf{j} + 1\mathbf{k}$.

The matrix representation is:

$$
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
2 & 1 & -3 \
4 & -2 & 1
\end{vmatrix}
$$

Calculating the determinant using cofactor expansion along the first row:

$\mathbf{a} \times \mathbf{b} = \mathbf{i} \begin{vmatrix} 1 & -3 \ -2 & 1 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & -3 \ 4 & 1 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & 1 \ 4 & -2 \end{vmatrix}$

$\mathbf{a} \times \mathbf{b} = \mathbf{i}((1)(1) - (-3)(-2)) - \mathbf{j}((2)(1) - (-3)(4)) + \mathbf{k}((2)(-2) - (1)(4))$

$\mathbf{a} \times \mathbf{b} = \mathbf{i}(1 - 6) - \mathbf{j}(2 - (-12)) + \mathbf{k}(-4 - 4)$

$\mathbf{a} \times \mathbf{b} = \mathbf{i}(-5) - \mathbf{j}(14) + \mathbf{k}(-8)$

$\mathbf{a} \times \mathbf{b} = -5\mathbf{i} - 14\mathbf{j} - 8\mathbf{k}$

This resulting vector is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.

Why Use This Method?

Using the determinant of this matrix is a common and organized way to remember and compute the components of the resulting cross product vector, especially for those familiar with matrix operations.

askvity