Matrix Operations

Cross Product of Two Vectors

Given two vectors:

\[\mathbf{A} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} d \\ e \\ f \end{bmatrix}\]

The cross product $\mathbf{A} \times \mathbf{B}$ is given by the determinant of a 3x3 matrix:

\[\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ d & e & f \end{vmatrix}\]

Components:

• X-component: \(\hat{i} \cdot \begin{vmatrix} b & c \\ e & f \end{vmatrix} = bf - ce\)

• Y-component: \(-\hat{j} \cdot \begin{vmatrix} a & c \\ d & f \end{vmatrix} = -(af - cd)\)

• Z-component: \(\hat{k} \cdot \begin{vmatrix} a & b \\ d & e \end{vmatrix} = ae - bd\)

Resulting Vector:

\[\mathbf{A} \times \mathbf{B} = \left( bf - ce, -(af - cd), ae - bd \right)\]

Determinant

\[A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\] \[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]