The cross product is a vector operation defined for three-dimensional vectors. Unlike the dot product, which produces a scalar,
the cross product produces another vector. If
\(A = \langle A_x, A_y, A_z \rangle\)
and
\(B = \langle B_x, B_y, B_z \rangle\),
then the cross product
\(A \times B\)
is perpendicular to both \(A\) and \(B\).
This makes it especially important in geometry, mechanics, torque, angular momentum, and the description of oriented areas.
Component formula
In component form, the cross product is computed from the determinant-style pattern
Cross-product formula.
\[
\begin{aligned}
A \times B
&= \left(A_yB_z - A_zB_y,\ A_zB_x - A_xB_z,\ A_xB_y - A_yB_x\right)
\end{aligned}
\]
So the three components are:
\[
\begin{aligned}
C_x &= A_yB_z - A_zB_y \\
C_y &= A_zB_x - A_xB_z \\
C_z &= A_xB_y - A_yB_x
\end{aligned}
\]
These are not arbitrary combinations. The alternating signs and cyclic structure encode the orientation of 3D space.
Because of that orientation,
\(A \times B\)
is not the same as
\(B \times A\).
Order matters.
\[
\begin{aligned}
B \times A &= -(A \times B)
\end{aligned}
\]
Reversing the order flips the direction of the result. This is one of the most important properties of the cross product.
Magnitude and area
The magnitude of the cross product is related to the sine of the angle between the vectors:
Magnitude formula.
\[
\begin{aligned}
|A \times B| &= |A|\,|B|\sin\theta
\end{aligned}
\]
This has a geometric meaning. If the vectors start from the same point, they span a parallelogram.
The magnitude
\(|A \times B|\)
equals the area of that parallelogram.
Therefore:
\[
\begin{aligned}
\text{Parallelogram area} &= |A \times B| \\
\text{Triangle area} &= \frac{1}{2}|A \times B|
\end{aligned}
\]
This immediately tells us two useful facts:
| Situation |
What happens to \(|A \times B|\) |
Why |
| Vectors are perpendicular |
Maximum for given lengths |
\(\sin 90^\circ = 1\) |
| Vectors are parallel or anti-parallel |
Zero |
\(\sin 0^\circ = \sin 180^\circ = 0\) |
| One vector is the zero vector |
Zero |
The spanned area disappears |
Direction and the right-hand rule
The direction of
\(A \times B\)
is determined by the right-hand rule. Point the fingers of your right hand in the direction of \(A\), then curl them toward \(B\).
Your thumb points in the direction of
\(A \times B\).
That direction is perpendicular to the plane containing \(A\) and \(B\).
This directional rule is why the cross product carries orientation information, not just magnitude.
It tells us which side of the plane is considered positive.
Important algebraic properties
| Property |
Formula |
Meaning |
| Anti-commutative |
\(A \times B = -(B \times A)\) |
Reversing the order flips the direction |
| Distributive |
\(A \times (B + C) = A \times B + A \times C\) |
The cross product distributes over addition |
| Scalar factor |
\((kA) \times B = k(A \times B)\) |
Scalars multiply the result magnitude |
| Self-cross |
\(A \times A = 0\) |
A vector forms zero area with itself |
Torque interpretation
In mechanics, torque is defined by
Torque formula.
\[
\begin{aligned}
\tau &= r \times F
\end{aligned}
\]
Here \(r\) is the position vector from the pivot point to the point where the force is applied, and \(F\) is the force vector.
The magnitude
\(|\tau| = |r||F|\sin\theta\)
measures how strongly the force tends to rotate the object.
The direction gives the axis of rotation through the right-hand rule.
This is why a force can have a large magnitude but produce small torque if it acts almost along the radius line.
Angular momentum interpretation
Angular momentum is defined by
Angular momentum formula.
\[
\begin{aligned}
L &= r \times p
\end{aligned}
\]
Here \(r\) is the position vector and \(p\) is the linear momentum.
Again, the cross product gives a vector perpendicular to the plane of motion.
This makes angular momentum an oriented rotational quantity, not just a number.
Worked example
Take the sample vectors
\(A = \langle 1,0,0 \rangle\)
and
\(B = \langle 0,1,0 \rangle\).
Step 1. Compute the components.
\[
\begin{aligned}
C_x &= A_yB_z - A_zB_y = 0\cdot 0 - 0\cdot 1 = 0 \\
C_y &= A_zB_x - A_xB_z = 0\cdot 0 - 1\cdot 0 = 0 \\
C_z &= A_xB_y - A_yB_x = 1\cdot 1 - 0\cdot 0 = 1
\end{aligned}
\]
Step 2. Combine the result.
\[
\begin{aligned}
A \times B &= \langle 0,0,1 \rangle
\end{aligned}
\]
Step 3. Compute the magnitude.
\[
\begin{aligned}
|A \times B| &= \sqrt{0^2 + 0^2 + 1^2} = 1
\end{aligned}
\]
So the parallelogram area is 1, the triangle area is \(1/2\), and the direction is \(+z\) by the right-hand rule.
Summary
| Concept |
Main formula |
Main meaning |
| Cross product |
\(A \times B = (A_yB_z - A_zB_y,\ A_zB_x - A_xB_z,\ A_xB_y - A_yB_x)\) |
Produces a vector perpendicular to both inputs |
| Magnitude |
\(|A \times B| = |A||B|\sin\theta\) |
Parallelogram area |
| Order |
\(B \times A = -(A \times B)\) |
Changing order flips the direction |
| Torque |
\(\tau = r \times F\) |
Rotational effect of a force |
| Angular momentum |
\(L = r \times p\) |
Oriented rotational momentum |