Key takeaways
- a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), a vector at right angles to both a and b.
- Its magnitude |a × b| is the area of the parallelogram made by a and b.
- Order matters: b × a points the opposite way to a × b.
- The calculator's main result is the magnitude. The vector itself is in the steps.
In this guide
What is the cross product?
Hold two pencils flat on a table at an angle. The cross product is a third arrow that points straight up out of the table, at right angles to both pencils. It is also called the vector product, because the answer is a vector.
Its magnitude (length) grows as the pencils get longer and as they open up towards a right angle.
Why does the cross product matter?
It is the go-to tool for anything that needs a direction perpendicular to two others. 3D games use it to light surfaces correctly, engineers use it for turning forces, and it gives areas in 3D without measuring any angles.
The vectors (2, 1, 0) and (1, 3, 0) lie flat on the floor. Their cross product is (0, 0, 5): it points straight up, and the parallelogram they make has an area of exactly 5.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a = (a₁, a₂, a₃) | The first vector | (2, 1, 0) |
| b = (b₁, b₂, b₃) | The second vector | (1, 3, 0) |
| i = a₂b₃ − a₃b₂ | First component of a × b | 0 |
| j = a₃b₁ − a₁b₃ | Second component | 0 |
| k = a₁b₂ − a₂b₁ | Third component | 6 − 1 = 5 |
| ‖a × b‖ | Magnitude of the cross product | 5 |
The magnitude also equals |a| × |b| × sin θ, where θ is the angle between the vectors. The calculator shows the cross product vector in its steps and gives its magnitude as the result.
How to calculate the cross product step by step
- 1First component.
i = a₂b₃ − a₃b₂. - 2Second component.
j = a₃b₁ − a₁b₃. - 3Third component.
k = a₁b₂ − a₂b₁. - 4Find the magnitude. Square i, j and k, add them, and take the square root. This is the vector magnitude of (i, j, k).
Now try your own two vectors in the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Assuming a × b = b × a. They point in opposite directions. The magnitude is the same.
- Confusing it with the [dot product](/calculator/dot-product), which gives a number, not a vector.
- Forgetting the square root when finding the magnitude.
Tips and tricks
- Triangle area: half the magnitude. The triangle made by (2, 1, 0) and (1, 3, 0) has area
5 ÷ 2 = 2.5. - Parallel vectors give a cross product of (0, 0, 0), because there is no area between them.
- Quick axis check: (1, 0, 0) × (0, 1, 0) = (0, 0, 1). x cross y gives z.
Where you'll use it in real life
- Geometry: the area of a parallelogram or triangle in 3D.
- Physics: torque, the turning effect of a force.
- 3D graphics: the normal (straight-out) direction of a surface for lighting.
- Electromagnetism: the force on a charge moving through a magnetic field.
Quick summary
The cross product of two 3D vectors is a new vector at right angles to both, and its magnitude is the area of their parallelogram. Follow the 2-3, 3-1, 1-2 cycle and keep the order. Use the calculator above to get the vector and its magnitude with every step shown.
Worked example
Two sides of a flat metal plate start at the same corner and are described by the vectors (2, 1, 0) and (1, 3, 0), in metres. The magnitude of their cross product gives the area of the parallelogram-shaped plate.
- 01a × b = (a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁)
- 02i = 1×0 - 0×3 = 0
- 03j = 0×1 - 2×0 = 0
- 04k = 2×3 - 1×1 = 5
- 05a × b = (0, 0, 5)
- 06|a × b| = √(25) = 5
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does the magnitude of the cross product mean?
It is the area of the parallelogram formed by the two vectors. Half of it is the area of the triangle they form.
When is the cross product zero?
When the two vectors are parallel (or one of them is zero). Then sin θ = 0, so there is no area between them.
Does the cross product work in 2D?
The true cross product needs 3D vectors. For 2D vectors, set the third components to 0; the answer then points straight along the z-axis.
