Cross Product Calculator – 3D Vector Cross Product & Area

Compute the cross product of two vectors in 2D or 3D. Get the resulting vector, its magnitude, and the area of the parallelogram and triangle spanned by the vectors, with clear formulas and steps.

Core math · Linear algebra · Vector calculus Updated:

Vector cross product calculator

Dimension

In 2D mode, z components are fixed to 0 and only the z component of a × b is reported.

Vector a

Vector b

Tip: enter unit vectors (e.g. i, j, k) or simple integer triples to check your hand calculations.

Resulting vector a × b

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Cartesian components

Magnitude |a × b|

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Length of cross product

Parallelogram area

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Area spanned by a and b

Triangle area

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Half of parallelogram area

Interpretation & notes

Enter two vectors to see the cross product and geometric interpretation.

Definition of the cross product

Given two vectors in three-dimensional space \[ \mathbf{a} = (a_x, a_y, a_z), \quad \mathbf{b} = (b_x, b_y, b_z), \] the cross product (or vector product) \(\mathbf{a} \times \mathbf{b}\) is a vector that:

  • is perpendicular (orthogonal) to both \(\mathbf{a}\) and \(\mathbf{b}\);
  • has magnitude \(|\mathbf{a}||\mathbf{b}|\sin\theta\), where \(\theta\) is the angle between them;
  • follows the right-hand rule for its direction.

Coordinate formula

\[ \mathbf{a} \times \mathbf{b} = \bigl(a_y b_z - a_z b_y,\;\; a_z b_x - a_x b_z,\;\; a_x b_y - a_y b_x\bigr). \]

Determinant mnemonic

\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}. \]

Geometric meaning: area and orientation

The magnitude of the cross product has a clear geometric interpretation:

\[ \bigl|\mathbf{a} \times \mathbf{b}\bigr| = |\mathbf{a}|\,|\mathbf{b}| \sin\theta. \]

  • \(|\mathbf{a} \times \mathbf{b}|\) equals the area of the parallelogram spanned by \(\mathbf{a}\) and \(\mathbf{b}\).
  • The area of the triangle with sides \(\mathbf{a}\) and \(\mathbf{b}\) is \(\dfrac{1}{2}|\mathbf{a} \times \mathbf{b}|\).

This is why the cross product is heavily used in physics and engineering to compute moments, torques, surface elements and oriented areas.

Cross product in 2D

In two dimensions we typically work with vectors of the form \(\mathbf{a} = (a_x, a_y)\) and \(\mathbf{b} = (b_x, b_y)\). A common convention is to interpret them as 3D vectors with zero z component: \[ \mathbf{a} = (a_x, a_y, 0), \quad \mathbf{b} = (b_x, b_y, 0). \]

In that case the cross product has only a z component: \[ \mathbf{a} \times \mathbf{b} = (0, 0, a_x b_y - a_y b_x). \] The scalar value \(a_x b_y - a_y b_x\) is often used as a signed area or orientation test in computational geometry.

Worked example

Consider the vectors \[ \mathbf{a} = (3, -2, 1), \quad \mathbf{b} = (1, 4, 0). \]

Using the coordinate formula:

\[ \mathbf{a} \times \mathbf{b} = \bigl( (-2)\cdot 0 - 1\cdot 4,\; 1\cdot 1 - 3\cdot 0,\; 3\cdot 4 - (-2)\cdot 1 \bigr) = (-4,\; 1,\; 14). \]

The magnitude is \[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(-4)^2 + 1^2 + 14^2} = \sqrt{16 + 1 + 196} = \sqrt{213} \approx 14.5945. \]

  • Area of the parallelogram spanned by \(\mathbf{a}\) and \(\mathbf{b}\): \(|\mathbf{a} \times \mathbf{b}| \approx 14.5945\).
  • Area of the corresponding triangle: \(\dfrac{1}{2}|\mathbf{a} \times \mathbf{b}| \approx 7.2973\).

If you enter the same numbers into the calculator with 4 decimal places selected, you should obtain numerically consistent results (up to rounding).

Common pitfalls and checks

  • Zero vector result. If \(\mathbf{a}\) and \(\mathbf{b}\) are parallel or one of them is the zero vector, then \(\mathbf{a} \times \mathbf{b} = \mathbf{0}\). The calculator will flag this in the notes.
  • Order matters. The cross product is anti-commutative: \[ \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}). \] Swapping the vectors flips the direction (and sign) of the result.
  • Units. If \(\mathbf{a}\) and \(\mathbf{b}\) carry physical units (e.g. meters, newtons), then the cross product inherits the product of those units (e.g. newton·meter for torque).

Where the cross product is used

  • Physics & engineering: torque, angular momentum, magnetic force.
  • Computer graphics: surface normals, lighting, back-face culling.
  • Geometry & CAD: oriented area, plane equations, intersection tests.

This calculator is intended as an educational and engineering support tool. For safety-critical or standards-bound work, always double-check results against your own derivations or trusted software and ensure the correct coordinate system and units.