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.

Geometric meaning: area and orientation

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

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:

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.