Dot Product, Cross Product, and Applications

Dot Product & Cross Product

The dot product measures alignment; the cross product measures perpendicularity. Picking the wrong one silently zeroes out your geometry.

Dot Product (scalar result): $$\vec{u}\cdot\vec{v} = u_xv_x + u_yv_y + u_zv_z$$ $$\vec{u}\cdot\vec{v} = \|\vec{u}\|\|\vec{v}\|\cos\theta$$

Cross Product (vector result, 3D only): $$\vec{u}\times\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix}$$

Common pitfall: The dot product peaks for parallel vectors and dies at 90 degrees; the cross product does the exact opposite.

Properties & Applications

Physics Applications:

• Work: $W = \vec{F}\cdot\vec{d}$
• Torque: $\vec{\tau} = \vec{r}\times\vec{F}$
• Magnetic Force: $\vec{F} = q\vec{v}\times\vec{B}$
• Angular Momentum: $\vec{L} = \vec{r}\times\vec{p}$