Cross Product (Vector Product)

The cross product turns two vectors into a third vector perpendicular to both. It encodes rotational quantities — torque, angular momentum, magnetic force.

Concept

Definition

For vectors $\vec{A}$ and $\vec{B}$ with angle $\theta$ between them: $\vec{A}\times\vec{B} = AB\sin\theta\,\hat{n},$ where $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{A}$ and $\vec{B}$, given by the right-hand rule.

Right-Hand Rule

Point your fingers along $\vec{A}$, curl toward $\vec{B}$; the thumb points along $\vec{A}\times\vec{B}$.

Component Form (Determinant)

$\vec{A}\times\vec{B} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\A_x&A_y&A_z\\B_x&B_y&B_z\end{vmatrix}$

Expanding: $\vec{A}\times\vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}.$

Identities of Unit Vectors

The cyclic pattern: $i \to j \to k \to i$ gives positive results.

Properties

Geometric Meaning

$|\vec{A}\times\vec{B}|$ = area of the parallelogram formed by $\vec{A}$ and $\vec{B}$.

Applications

• Torque: $\vec{\tau} = \vec{r}\times\vec{F}$.
• Angular momentum: $\vec{L} = \vec{r}\times\vec{p}$.
• Magnetic force on a charge: $\vec{F} = q\vec{v}\times\vec{B}$.
• Area vector for a planar loop.

Worked Example

Q1: $\vec{A} = 2\hat{i} + 3\hat{j}$, $\vec{B} = \hat{i} - \hat{j} + 2\hat{k}$. Find $\vec{A}\times\vec{B}$.

Solution: $\vec{A}\times\vec{B} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&0\\1&-1&2\end{vmatrix}$

• $\hat{i}$-component: $3(2) - 0(-1) = 6$.
• $\hat{j}$-component: $-(2(2) - 0(1)) = -4$.
• $\hat{k}$-component: $2(-1) - 3(1) = -5$.

So $\vec{A}\times\vec{B} = 6\hat{i} - 4\hat{j} - 5\hat{k}$.

Q2: Torque from $\vec{F} = 3\hat{i}$ N applied at $\vec{r} = 2\hat{j}$ m from origin: $\vec{\tau} = \vec{r}\times\vec{F} = 2\hat{j}\times 3\hat{i} = 6(\hat{j}\times\hat{i}) = -6\hat{k}\,\text{N·m}.$

Common Confusions

• "$\vec{A}\times\vec{B} = \vec{B}\times\vec{A}$." — Wrong, it changes sign: anti-commutative.
• "$\vec{A}\times\vec{B} = 0$ implies $\vec{A} = \vec{0}$." — Could also mean $\vec{A}\parallel\vec{B}$.
• "Cross product gives a scalar." — No, it gives a vector.
• Wrong sign on $\hat{j}$ in the determinant expansion — always remember the alternating signs $+, -, +$.

Key Takeaways

• $\vec{A}\times\vec{B} = AB\sin\theta\,\hat{n}$, direction by right-hand rule.
• Magnitude equals the area of the parallelogram spanned by the two vectors.
• Anti-commutative; distributive; self-cross is zero.
• Computed via the $3\times 3$ determinant in components.
• Cornerstone of rotational and magnetic physics.