The cross product takes two vectors in three-dimensional space and produces a third vector that is perpendicular to both. Its length equals the area of the parallelogram the two vectors span, and its direction follows the right-hand rule. That makes it the standard tool for finding normal vectors to planes, torque in physics and areas in 3D geometry. This calculator does the determinant arithmetic and reports the vector, its length, a unit normal and the angle between the inputs.
How to use the cross product calculator
- Type vector a as three components separated by commas, such as
2, 3, 4. Fractions and decimals are allowed. - Type vector b the same way. Two-component vectors are accepted and treated as lying in the xy-plane.
- Read a × b on the tape, along with the parallelogram and triangle areas, the unit normal, the angle and b × a. The steps expand the determinant one component at a time and check that the result is perpendicular to both inputs.
Cross product formula
A memory aid: write the vectors as the second and third rows of a 3 × 3 determinant whose first row is i, j, k, then expand along the first row. Each component is a 2 × 2 determinant made by covering that component’s column.
Worked example
Let a = ⟨2, 3, 4⟩ and b = ⟨5, 6, 7⟩.
i: 3·7 − 4·6 = 21 − 24 = −3
j: 4·5 − 2·7 = 20 − 14 = 6
k: 2·6 − 3·5 = 12 − 15 = −3
Result: a × b = ⟨−3, 6, −3⟩.
Check: ⟨−3, 6, −3⟩ · ⟨2, 3, 4⟩ = −6 + 18 − 12 = 0 and ⟨−3, 6, −3⟩ · ⟨5, 6, 7⟩ = −15 + 36 − 21 = 0, so the result is perpendicular to both.
Area: |a × b| = √(9 + 36 + 9) = √54 = 3√6 ≈ 7.348, so the triangle with these two sides has area ≈ 3.674.
The two vectors are close to parallel (the angle between them is only about 7.48°), which is why the parallelogram is thin compared with the vectors’ lengths of about 5.39 and 10.49.
Properties
| Property | Statement |
|---|---|
| Anti-commutative | b × a = −(a × b) |
| Perpendicular | (a × b) · a = (a × b) · b = 0 |
| Parallel test | a × b = 0 exactly when a and b are parallel |
| Distributive | a × (b + c) = a × b + a × c |
| Not associative | (a × b) × c is usually not a × (b × c) |
| Unit vectors | i × j = k, j × k = i, k × i = j |
The right-hand rule
Point the fingers of your right hand along a and curl them toward b; your thumb points in the direction of a × b. Swapping the order reverses the curl and the thumb flips, which is the anti-commutative property in physical form.
Applications
- Normal vector to a plane. Through points P, Q and R, the vectors PQ and PR lie in the plane, and PQ × PR is perpendicular to it. Its components are the coefficients A, B, C in the plane equation Ax + By + Cz = D.
- Area of a 3D triangle. Half the magnitude of PQ × PR gives the area directly, without needing side lengths and angles.
- Torque. A force F applied at position r from a pivot creates torque τ = r × F. Its magnitude rF sin θ explains why pushing a door at the handle, perpendicular to it, is easiest.
- Magnetic force. A charge q moving with velocity v in a magnetic field B feels F = q(v × B), always at right angles to its motion.
For the scalar companion operation and the angle formula based on cosine, see the dot product calculator. To find the distance between two points in space, use the 3D distance calculator.
Frequently asked questions
How do you calculate a cross product?
For a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩, a × b = ⟨a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁⟩. The easiest way to remember it is as a 3 × 3 determinant with i, j and k in the top row.
What does the magnitude of the cross product mean?
|a × b| equals the area of the parallelogram with sides a and b, and half of it is the area of the triangle they form. It also equals |a| |b| sin θ, where θ is the angle between the vectors.
Why is a × b different from b × a?
The cross product is anti-commutative: b × a = −(a × b). Both are perpendicular to a and b, but they point in opposite directions, as the right-hand rule shows.
What does a zero cross product mean?
The vectors are parallel (one is a scalar multiple of the other) or at least one of them is the zero vector. Parallel vectors span no area, so the parallelogram collapses to a line.
Can I take the cross product of 2D vectors?
Treat them as 3D vectors with z = 0. The result then points straight along the z-axis, and its z-component a₁b₂ − a₂b₁ is the signed area of the parallelogram in the plane.