What's vector product with mathematical analysis

Mathematically, if $\vec{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$ and $\vec{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}$, their cross product is calculated as the determinant:

$$\vec{A} \times \vec{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k}$$

In mathematical analysis, the vector product is fundamental in vector calculus for defining quantities such as torque and angular momentum, and it is essential for differentiating vector functions, where the product rule for cross products applies: $\frac{d}{du}(\vec{A} \times \vec{B}) = \vec{A} \times \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} \times \vec{B}$.