Cross product
Produce a vector perpendicular to two three-dimensional vectors.
Cross product is one of 5 vectors formulas in the linear algebra section of this library, and it is used at university level.
Why cross product works
Each component of the answer is a small 2 by 2 determinant built from the other two coordinates, which measures the signed area of the shadow that u and v cast on that coordinate plane. Because each component leaves its own axis out of the calculation, the result automatically dots to zero with both inputs, which is exactly what perpendicular means.
What each symbol means
$\mathbf u,\mathbf v\in\mathbb R^3$.
Cross product: when it holds
Order matters: $\mathbf v\times\mathbf u=-(\mathbf u\times\mathbf v)$.
When it stops applying
If the two vectors are parallel, they span no area, and the result is the zero vector with no direction at all. Try (1, 2, 3) × (2, 4, 6): every component cancels to 0, because the second vector is just double the first.
Cross product: a worked example
$(1,0,0)\times(0,1,0)=(0,0,1)$.
The mistake to avoid
What people do: Students subtract in the wrong order inside a component, or swap the two vectors and keep the same answer.
Why it goes wrong: Reversing the order reverses the result: (1, 0, 0) × (0, 1, 0) is (0, 0, 1), while (0, 1, 0) × (1, 0, 0) is (0, 0, −1). The arrow flips to the other side of the plane.
Do this instead: Fix an order and keep it, then check with the right-hand rule: point your fingers along the first vector, curl them toward the second, and your thumb shows which way the answer points.
Cross product: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\mathbf u,\mathbf v\in\mathbb R^3$.
- Check the conditions before substituting. Order matters: $\mathbf v\times\mathbf u=-(\mathbf u\times\mathbf v)$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most linear algebra slips.
Where this formula fits
- Subject
- Linear Algebra formulas — 17 entries in this library
- Topic
- Vectors
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where cross product comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Dot Product — the lesson behind this formula: measure alignment and test orthogonality.
- Normal Vector Calculator — check your substitution and the value it produces.
- Study linear algebra — the subject guide that explains the ideas these formulas compress.
- Linear Algebra Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 linear algebra formulas — the full grouped reference, or the complete formula library.
Questions about cross product
Does the cross product exist for 2D vectors?
Not as a vector. In the plane there is no third direction to point in, so people either use the scalar u₁v₂ − u₂v₁ as a signed area or pad both vectors with a zero third component and read off the z-value.
What does the length of the result tell me?
It equals ‖u‖‖v‖ sin θ, the area of the parallelogram the two vectors span. Half of that is the area of the triangle they form, which makes this a fast way to get areas in 3D.
How is it different from the dot product?
The dot product returns a number that peaks when the vectors line up, while the cross product returns a vector that peaks when they are perpendicular. One measures agreement, the other measures spread.
Is it associative like ordinary multiplication?
No. In general u × (v × w) is not (u × v) × w, so the parentheses matter. There is a separate expansion rule for the triple product when you actually need to unpack one.