Cross Product Solver

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Compute the cross product \(\mathbf{A}\times\mathbf{B}\) in 3D, its magnitude \(\lVert\mathbf{A}\times\mathbf{B}\rVert\), the angle \(\theta\) (via \(\sin\theta\) / \(\cos\theta\)), and the right-hand rule direction (unit normal). The magnitude equals the area of the parallelogram spanned by \(\mathbf{A}\) and \(\mathbf{B}\).

3D cross product visualization

3D: drag to rotate • Shift+drag to pan • wheel/trackpad to zoom • double-click to reset. “Play” rotates \(\mathbf{B}\) once (360°) and stops automatically.

\(\mathbf{A}\) \(\mathbf{B}\) \(\mathbf{A}\times\mathbf{B}\) angle arc parallelogram (dashed)

Enter vectors and click “Calculate”.

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Theory

Cross product in \(\mathbb{R}^3\)

The cross product takes two 3D vectors and returns a 3D vector perpendicular to both: \[ \mathbf{A}\times\mathbf{B}= \begin{bmatrix} a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1 \end{bmatrix}. \]

A compact way to remember it is the determinant: \[ \mathbf{A}\times\mathbf{B}= \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_1&a_2&a_3\\ b_1&b_2&b_3 \end{vmatrix}. \]

Magnitude and area

The magnitude satisfies \[ \lVert\mathbf{A}\times\mathbf{B}\rVert=\lVert\mathbf{A}\rVert\,\lVert\mathbf{B}\rVert\sin\theta, \] where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).

\(\lVert\mathbf{A}\times\mathbf{B}\rVert\) is the area of the parallelogram spanned by \(\mathbf{A}\) and \(\mathbf{B}\).
\(\tfrac12\lVert\mathbf{A}\times\mathbf{B}\rVert\) is the area of the triangle spanned by them.

Right-hand rule and direction

The direction of \(\mathbf{A}\times\mathbf{B}\) is determined by the right-hand rule: curl your fingers from \(\mathbf{A}\) toward \(\mathbf{B}\), and your thumb points along \(\mathbf{A}\times\mathbf{B}\).

A unit normal vector is \[ \hat{\mathbf{n}}=\frac{\mathbf{A}\times\mathbf{B}}{\lVert\mathbf{A}\times\mathbf{B}\rVert}, \] which is defined only when \(\mathbf{A}\times\mathbf{B}\neq \mathbf{0}\).

Parallel check

Two vectors are parallel (or anti-parallel) exactly when their cross product is zero: \[ \mathbf{A}\times\mathbf{B}=\mathbf{0} \quad\Longleftrightarrow\quad \sin\theta=0 \quad\Longleftrightarrow\quad \theta\in\{0^\circ,180^\circ\}. \]

Physics example: torque

Torque is a cross product: \[ \boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}. \] The magnitude \(\lVert\boldsymbol{\tau}\rVert=\lVert\mathbf{r}\rVert\,\lVert\mathbf{F}\rVert\sin\theta\) measures how strongly a force causes rotation about a pivot.

University note: wedge product

In advanced algebra, the cross product is related to the exterior (wedge) product, which generalizes “oriented area/volume” beyond \(\mathbb{R}^3\). The usual 3D cross product is a special case tied to 3D geometry.

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