How do you calculate the cross product of two vectors?

The cross product of two vectors is calculated using the determinant method, which provides a systematic way of finding the resultant vector that is perpendicular to the plane formed by the two input vectors. This method employs a determinant of a matrix that includes the unit vectors i, j, and k in the first row, the components of the first vector in the second row, and the components of the second vector in the third row.

When calculating the cross product using the determinant, you set up the matrix as follows:

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

Here, the unit vectors represent the x, y, and z-axis directions, while (a_1, a_2, a_3) and (b_1, b_2, b_3) represent the components of the first and second vectors, respectively. The cross product is computed by expanding this determinant, resulting in a new vector that embodies both the magnitude and direction determined by the original two vectors.

This method emphasizes the geometric