Answer
See the explanation
Work Step by Step
The determinant formula for calculating the cross product of two vectors in the Cartesian coordinate system is as follows:
\[
\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}
\]
where \(\mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}\) and \(\mathbf{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k}\) are the given vectors.
To illustrate this formula, let's consider the following example:
\(\mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{B} = -1\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}\).
Using the determinant formula, we can calculate the cross product as follows:
\[
\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -3 & 4 \\ -1 & 5 & 2 \end{vmatrix}
\]
Expanding the determinant, we have:
\[
\mathbf{A} \times \mathbf{B} = ( -3 \cdot 2 - 5 \cdot 4 )\mathbf{i} - ( 2 \cdot 2 - 5 \cdot -1 )\mathbf{j} + ( 2 \cdot 4 - ( -3 \cdot -1 ))\mathbf{k}
\]
Simplifying further, we get:
\[
\mathbf{A} \times \mathbf{B} = -22\mathbf{i} - 9\mathbf{j} + 5\mathbf{k}
\]
Therefore, the cross product of vectors \(\mathbf{A}\) and \(\mathbf{B}\) is \(-22\mathbf{i} - 9\mathbf{j} + 5\mathbf{k}\).
