Dot Product, Cross Product, and Applications

Dot product (scalar result):

$$\vec{u}\cdot\vec{v} = u_xv_x + u_yv_y + u_zv_z = \|\vec{u}\|\,\|\vec{v}\|\cos\theta$$

Cross product (vector result, 3D only):

$$\vec{u}\times\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{vmatrix}$$

Comparison

• Dot → scalar; cross → vector
• Dot is commutative; cross is anti-commutative
• Dot measures projection; cross measures area
• $Dot = 0 when \perp$; cross = $\vec{0}$ when $\parallel$

Applications in physics

• 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: $\vec{L} = \vec{r}\times\vec{p}$

Key insight: The dot product measures alignment; the cross product measures perpendicularity.

Common pitfall: The dot product peaks for parallel vectors and dies at 90°; the cross product does the exact opposite. Picking the wrong product silently zeroes out the very geometry you meant to measure.