The cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the plane formed by the original vectors. Unlike the dot product, which results in a scalar, the cross product yields a vector. This operation is particularly useful in physics and engineering, where understanding the orientation of vectors is crucial.

To compute the cross product of two vectors, say \(\mathbf{u}\) and \(\mathbf{v}\), we can follow a systematic approach. For example, consider vectors \(\mathbf{u} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix}\). The first step involves setting up a matrix with the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) in the first row, followed by the components of \(\mathbf{u}\) and \(\mathbf{v}\) in the subsequent rows:

\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\2 & 0 & 1 \\0 & -1 & 2\end{vmatrix}\end{equation}

Next, we repeat the first two rows of unit vectors to facilitate the calculation:

\[\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\2 & 0 & 1 \\0 & -1 & 2 \\\mathbf{i} & \mathbf{j} & \mathbf{k}\end{vmatrix}\end{equation}

To find the components of the resulting vector, we apply the cross product formula, which involves calculating the determinant of the matrix. The components are derived using the pattern of multiplying the components of the vectors in a cross-down and up manner:

For the \(\mathbf{i}\) component (x-component), we calculate:

\(0 \cdot 2 - (-1) \cdot 1 = 0 + 1 = 1\)

For the \(\mathbf{j}\) component (y-component), we compute:

\(1 \cdot 0 - 2 \cdot 2 = 0 - 4 = -4\)

For the \(\mathbf{k}\) component (z-component), we find:

\(2 \cdot (-1) - 0 \cdot 0 = -2 - 0 = -2\)

Thus, the resulting vector from the cross product \(\mathbf{w} = \mathbf{u} \times \mathbf{v}\) is:

\(\mathbf{w} = \begin{pmatrix} 1 \\ -4 \\ -2 \end{pmatrix}\)

In summary, the cross product not only provides a vector that is orthogonal to the original vectors but also follows a consistent pattern of multiplying the unlike components. This methodical approach simplifies the process, making it easier to compute the cross product in various applications.