Cross Product

Definition

The cross product (or vector product) of two vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^3$ is a vector that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ , with a magnitude proportional to the area of the parallelogram they span.

Why It Matters

The cross product is essential for modeling rotation, torque, and electromagnetism in three-dimensional space. It provides a mathematical bridge between area, orientation, and orthogonal force, which is fundamental to physics and engineering.

Core Concepts

Geometric Definition: $\mathbf{u} \times \mathbf{v} = (|\mathbf{u}| |\mathbf{v}| \sin \theta) \mathbf{n}$ $u \times v = (∣ u ∣∣ v ∣ sin θ) n$ , where $\mathbf{n}$ $n$ is a unit vector perpendicular to the plane of $\mathbf{u}$ $u$ and $\mathbf{v}$ $v$ , determined by the right-hand rule.
- How to read: “The cross product of u and v equals the magnitude of u times the magnitude of v times the sine of theta, times the unit normal vector n.”
- Meaning: Magnitude equals parallelogram area; $\mathbf{n}$ is the unit normal perpendicular to $\mathbf{u}$ and $\mathbf{v}$ (right-hand rule).
Determinant Formula: $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$ $u \times v = i u_{1} v_{1} j u_{2} v_{2} k u_{3} v_{3}$ .
- How to read: “The cross product of u and v equals the three by three determinant with unit vectors i, j, k, followed by the components of u and v.”
- Meaning / when to use: Expand along the first row for component-wise computation.
Properties: It is anticommutative ( $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$ $u \times v = - (v \times u)$ ) and yields the zero vector if the input vectors are parallel.
- How to read: “The cross product of u and v equals the negative cross product of v and u.”
- Meaning: Swapping order flips the direction; parallel vectors give zero cross product.
Area of Parallelogram: The magnitude $|\mathbf{u} \times \mathbf{v}|$ is exactly the area of the parallelogram determined by the two vectors.

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes