Vector Cross Product Calculator (3D)

This calculator works out the cross product a × b of two vectors in three dimensions. Enter the three components of a and of b; you get the x, y and z components of a × b and its length, which equals the area of the parallelogram spanned by a and b.

Formula

a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), and |a × b| = |a| |b| sin θ, where θ is the angle between a and b.

• a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃)
• The result is perpendicular to both a and b, with its direction given by the right-hand rule.

Worked Example

Take a = (1, 2, 3) and b = (4, 5, 6).

1. x: 2 × 6 − 3 × 5 = 12 − 15 = -3.000
2. y: 3 × 4 − 1 × 6 = 12 − 6 = 6.000
3. z: 1 × 5 − 2 × 4 = 5 − 8 = -3.000
4. Length = √(9 + 36 + 9) = √54 = 7.348

How to read the result

Use the cross product to find a normal vector to a plane through three points (cross two edge vectors), the area of a triangle (half the length), or torque (r × F). A zero result means the vectors are parallel.

Limits and common mistakes

• Order matters: b × a = −(a × b).
• The cross product exists only in three dimensions (and seven). For 2D vectors, set both z components to 0; the answer is then (0, 0, a₁b₂ − a₂b₁).
• Do not confuse it with the dot product, which gives a single number.

Frequently Asked Questions

What does a cross product of zero mean?

The two vectors are parallel (or one of them is zero).

How do I get the area of a triangle from two sides?

Half the length of the cross product of the two side vectors.

Is the cross product commutative?

No. Swapping the vectors reverses the sign.

What is i × j?

k, with i, j and k the unit vectors along x, y and z.