Cross product calculator

What is a cross product calculator?

A cross product calculator finds the vector that results from multiplying two three-dimensional vectors together using the cross (or vector) product. Unlike the dot product, which returns a single number, the cross product returns a new vector. That vector is perpendicular to both of the original vectors and its length equals the area of the parallelogram they span.

Given two vectors $\mathbf{a} = (a_x, a_y, a_z)$ and $\mathbf{b} = (b_x, b_y, b_z)$, this tool returns the three components of $\mathbf{c} = \mathbf{a} \times \mathbf{b}$.

Formula

The cross product is defined component by component as:

So the three output components are:

• $c_x = a_y b_z - a_z b_y$
• $c_y = a_z b_x - a_x b_z$
• $c_z = a_x b_y - a_y b_x$

How to use

1. Enter the three components of vector $\mathbf{a}$: $a_x$, $a_y$, and $a_z$.
2. Enter the three components of vector $\mathbf{b}$: $b_x$, $b_y$, and $b_z$.
3. Once all six values are filled in, the calculator displays $c_x$, $c_y$, and $c_z$ — the components of the resulting vector $\mathbf{a} \times \mathbf{b}$.

Negative inputs are fully supported. The order matters: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$, so swapping the two vectors flips the sign of every component.

Worked example

Take $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, 5, 6)$.

• $c_x = a_y b_z - a_z b_y = 2 \cdot 6 - 3 \cdot 5 = 12 - 15 = -3$
• $c_y = a_z b_x - a_x b_z = 3 \cdot 4 - 1 \cdot 6 = 12 - 6 = 6$
• $c_z = a_x b_y - a_y b_x = 1 \cdot 5 - 2 \cdot 4 = 5 - 8 = -3$

So $\mathbf{a} \times \mathbf{b} = (-3, 6, -3)$.

FAQ

Why is the cross product a vector while the dot product is a number?

The dot product measures how much two vectors point in the same direction, which is a single scalar quantity. The cross product instead measures the oriented area they span and points in a direction perpendicular to both, so it naturally needs three components to describe both that magnitude and that direction.

What does it mean if the cross product is the zero vector?

If $\mathbf{a} \times \mathbf{b} = (0, 0, 0)$, the two vectors are parallel (or one of them is the zero vector). Parallel vectors span no area, so the perpendicular result collapses to nothing.