Cross Product Calculator

Calculate the cross product of two 3D vectors with the determinant formula. Get a step-by-step expansion, the perpendicular result vector, its magnitude as the parallelogram area, a direction check and a 3D visualization.

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About Cross Product Calculator

The Cross Product Calculator computes the vector product of two 3D vectors using the determinant formula. Enter the components of two vectors to instantly get the resulting perpendicular vector, its magnitude (parallelogram area), the angle between the input vectors, step-by-step determinant expansion, perpendicularity verification, and an interactive 3D diagram you can rotate by dragging.

The Cross Product Formula

The cross product of two 3D vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$ is defined as the determinant:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$

Expanding by cofactors along the first row gives:

$$\vec{a} \times \vec{b} = \hat{i}(a_2 b_3 - a_3 b_2) - \hat{j}(a_1 b_3 - a_3 b_1) + \hat{k}(a_1 b_2 - a_2 b_1)$$

Real-World Applications

🔧

Torque

τ = r × F calculates rotational force

🌊

Angular Momentum

L = r × p in rotational mechanics

🎮

Surface Normals

3D rendering uses normals for lighting

✈

Electromagnetism

F = qv × B for Lorentz force

📐

Area Calculation

|a×b| gives parallelogram/triangle area

🏗

Structural Engineering

Moment of forces about a point

Key Formulas

Property	Formula	Description
Cross Product	$\vec{a} \times \vec{b} = \langle a_2 b_3 - a_3 b_2,\; a_3 b_1 - a_1 b_3,\; a_1 b_2 - a_2 b_1 \rangle$	Component form of the cross product
Magnitude	$\|\vec{a} \times \vec{b}\| = \|\vec{a}\|\|\vec{b}\|\sin\theta$	Equals the parallelogram area
Anti-commutativity	$\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$	Swapping order reverses the direction
Perpendicularity	$(\vec{a} \times \vec{b}) \cdot \vec{a} = 0$	Result is always perpendicular to both inputs
Parallel test	$\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \\| \vec{b}$	Zero cross product means vectors are parallel
Triangle area	$A = \frac{1}{2}\|\vec{a} \times \vec{b}\|$	Half the parallelogram area

Cross Product vs. Dot Product

Cross Product (a × b)

Produces a vector perpendicular to both inputs. Only defined in 3D. Magnitude equals the parallelogram area. Zero when vectors are parallel. Maximum when vectors are perpendicular. Anti-commutative: a × b = -(b × a).

Dot Product (a · b)

Produces a scalar value. Works in any dimension. Measures alignment between vectors. Zero when vectors are perpendicular. Maximum when vectors are parallel. Commutative: a · b = b · a.

Key Properties

Anti-commutative

$\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$ — order matters

Distributive

$\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$

Scalar Factor

$(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})$

Self Cross

$\vec{a} \times \vec{a} = \vec{0}$ — always zero

Not Associative

$\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$

Lagrange Identity

$|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2$

Understanding the Right-Hand Rule

The direction of the cross product follows the right-hand rule: point your right hand's fingers along the first vector $\vec{a}$, curl them toward the second vector $\vec{b}$, and your thumb indicates the direction of $\vec{a} \times \vec{b}$. This is why the cross product is anti-commutative — reversing the order reverses the thumb direction, giving $\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})$.

The magnitude $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ represents the area of the parallelogram formed by the two vectors. When the vectors are parallel ($\theta = 0°$ or $180°$), the area collapses to zero. When they are perpendicular ($\theta = 90°$), the area is maximized at $|\vec{a}| \times |\vec{b}|$.

How to Use the Cross Product Calculator

Enter Vector a: Type the three components (x, y, z) separated by commas — for example, 2, 3, 4. You can also click a quick example to auto-fill both vectors.
Enter Vector b: Type the three components of the second vector in the same format.
Watch the live preview: The 3D preview updates in real time, showing both vectors, the cross product vector, and the parallelogram.
Click Calculate: Press the button to get full results including the perpendicular result vector, parallelogram area, angle, step-by-step determinant expansion, and the interactive 3D diagram.
Explore the diagram: Drag to rotate the 3D view, toggle layers (parallelogram, cross product vector, axes, labels) for different visualizations.

FAQ

What is the cross product of two vectors?

The cross product (also called vector product) of two 3D vectors a and b produces a new vector that is perpendicular to both a and b. It is computed using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row. The magnitude of the result equals the area of the parallelogram formed by the two vectors.

How do you calculate the cross product using the determinant method?

Set up a 3×3 matrix with i-hat, j-hat, k-hat in row 1, components of vector a in row 2, and components of vector b in row 3. Then expand along the first row: i(a₂b₃ − a₃b₂) − j(a₁b₃ − a₃b₁) + k(a₁b₂ − a₂b₁). This gives the three components of the cross product vector.

Why is the cross product only defined in 3D?

The cross product as a vector operation is only defined in 3 dimensions (and 7 dimensions) because only in these spaces can you find a unique vector perpendicular to two given vectors. In 2D there is no out-of-plane direction, and in higher dimensions the perpendicular space has more than one dimension.

What is the geometric meaning of the cross product magnitude?

The magnitude |a × b| equals the area of the parallelogram formed by vectors a and b. Half this value gives the triangle area. This is widely used in physics (torque = r × F) and computer graphics (calculating surface normals for lighting).

What is the right-hand rule for cross products?

The right-hand rule determines the direction of the cross product: point your fingers along vector a, curl them toward vector b, and your thumb points in the direction of a × b. This means the cross product is anti-commutative — a × b equals −(b × a).

Reference this content, page, or tool as:

"Cross Product Calculator" at https://MiniWebtool.com/cross-product-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-10

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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