KAHIBARO
Table of Contents
Meaning of the Cross Product
The cross product is a way to combine two vectors and produce a new vector that is perpendicular to both of them. It is used when direction matters in a special way, especially in rotational physics, magnetism, and geometry.
Unlike the dot product, which gives a scalar, the cross product gives a vector. If two vectors are called $\vec{A}$ and $\vec{B}$, then their cross product is written as
$$
\vec{A} \times \vec{B}
$$
The result points in a direction that is at right angles to the plane containing $\vec{A}$ and $\vec{B}$.
Magnitude of the Cross Product
The size, or magnitude, of the cross product depends on the magnitudes of the two vectors and the angle between them:
$$
|\vec{A} \times \vec{B}| = AB \sin\theta
$$
where $A = |\vec{A}|$, $B = |\vec{B}|$, and $\theta$ is the angle between the vectors.
This formula tells us several important things. If the vectors are parallel, then $\theta = 0^\circ$ or $180^\circ$, so $\sin\theta = 0$, and the cross product is zero. If the vectors are perpendicular, then $\theta = 90^\circ$, so $\sin\theta = 1$, and the cross product has its maximum magnitude.
Important formula:
$$
|\vec{A} \times \vec{B}| = AB \sin\theta
$$
The cross product is zero for parallel vectors and largest for perpendicular vectors.
Direction of the Cross Product
The direction of $\vec{A} \times \vec{B}$ is found using the right hand rule. Point the fingers of your right hand in the direction of $\vec{A}$, then curl them toward $\vec{B}$. Your thumb points in the direction of $\vec{A} \times \vec{B}$.
This direction is perpendicular to the plane of the two vectors.
If you reverse the order of the vectors, the direction reverses:
$$
\vec{B} \times \vec{A} = -(\vec{A} \times \vec{B})
$$
So the cross product is not commutative.
Order matters:
$$
\vec{A} \times \vec{B} \neq \vec{B} \times \vec{A}
$$
In fact,
$$
\vec{B} \times \vec{A} = -(\vec{A} \times \vec{B})
$$
Geometric Interpretation
The magnitude of the cross product also has a geometric meaning. It equals the area of the parallelogram formed by the two vectors.
$$
\text{Area of parallelogram} = |\vec{A} \times \vec{B}|
$$
Since a triangle is half of a parallelogram, the area of the triangle formed by the two vectors is
$$
\text{Area of triangle} = \frac{1}{2} |\vec{A} \times \vec{B}|
$$
This interpretation is very useful in geometry and physics.
Cross Product in Component Form
In three dimensions, if
$$
\vec{A} = (A_x, A_y, A_z), \qquad \vec{B} = (B_x, B_y, B_z)
$$
then the cross product is
$$
\vec{A} \times \vec{B}
=
\left(
A_y B_z - A_z B_y,\,
A_z B_x - A_x B_z,\,
A_x B_y - A_y B_x
\right)
$$
This formula is often the easiest way to calculate the cross product numerically.
It can also be written using a determinant-like form:
$$
\vec{A} \times \vec{B}
=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix}
$$
This expands to the same component formula above.
For vectors
$$
\vec{A} = (A_x, A_y, A_z), \qquad \vec{B} = (B_x, B_y, B_z)
$$
the cross product is
$$
\vec{A} \times \vec{B}
=
\left(
A_y B_z - A_z B_y,\,
A_z B_x - A_x B_z,\,
A_x B_y - A_y B_x
\right)
$$
Cross Products of Unit Vectors
The standard unit vectors follow a simple pattern:
$$
\hat{i} \times \hat{j} = \hat{k}
$$
$$
\hat{j} \times \hat{k} = \hat{i}
$$
$$
\hat{k} \times \hat{i} = \hat{j}
$$
Reversing the order changes the sign:
$$
\hat{j} \times \hat{i} = -\hat{k}
$$
$$
\hat{k} \times \hat{j} = -\hat{i}
$$
$$
\hat{i} \times \hat{k} = -\hat{j}
$$
Also, any unit vector crossed with itself is zero:
$$
\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0
$$
| Cross product | Result |
|---|---|
| $\hat{i} \times \hat{j}$ | $\hat{k}$ |
| $\hat{j} \times \hat{k}$ | $\hat{i}$ |
| $\hat{k} \times \hat{i}$ | $\hat{j}$ |
| $\hat{j} \times \hat{i}$ | $-\hat{k}$ |
| $\hat{k} \times \hat{j}$ | $-\hat{i}$ |
| $\hat{i} \times \hat{k}$ | $-\hat{j}$ |
Important Properties
The cross product has several useful algebraic properties. It is anti commutative:
$$
\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}
$$
It distributes over addition:
$$
\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}
$$
and
$$
(\vec{A} + \vec{B}) \times \vec{C} = \vec{A} \times \vec{C} + \vec{B} \times \vec{C}
$$
If a vector is multiplied by a scalar $c$, then
$$
(c\vec{A}) \times \vec{B} = c(\vec{A} \times \vec{B})
$$
and
$$
\vec{A} \times (c\vec{B}) = c(\vec{A} \times \vec{B})
$$
A Simple Example
Take
$$
\vec{A} = (1,2,0), \qquad \vec{B} = (3,0,0)
$$
Then
$$
\vec{A} \times \vec{B}
=
\left(
2\cdot 0 - 0\cdot 0,\,
0\cdot 3 - 1\cdot 0,\,
1\cdot 0 - 2\cdot 3
\right)
=
(0,0,-6)
$$
So the result points in the negative $z$ direction.
Its magnitude is
$$
|\vec{A} \times \vec{B}| = 6
$$
This means the parallelogram formed by $\vec{A}$ and $\vec{B}$ has area $6$.
Why the Cross Product Matters in Physics
The cross product appears whenever a physical quantity depends on both a direction in space and a turning effect or perpendicular relationship.
For example, torque is defined using a cross product:
$$
\vec{\tau} = \vec{r} \times \vec{F}
$$
This tells us that torque depends on the position vector and the applied force, and that the torque direction is perpendicular to the plane of $\vec{r}$ and $\vec{F}$.
Another important example is magnetic force on a moving charge:
$$
\vec{F} = q \vec{v} \times \vec{B}
$$
This shows that the magnetic force is perpendicular to both the velocity and the magnetic field.
These topics are developed in later chapters, but they show why the cross product is such an important mathematical tool in physics.
Visualizing the Direction
It is common to draw perpendicular directions using symbols. A dot means a vector coming out of the page, and a cross means a vector going into the page.
In this picture, $\vec{A}$ points to the right and $\vec{B}$ points upward. Using the right hand rule, $\vec{A} \times \vec{B}$ points out of the page.
Common Mistakes
A common mistake is to use $\cos\theta$ instead of $\sin\theta$. The cross product uses $\sin\theta$, not $\cos\theta$.
Another common mistake is forgetting that order matters. Switching the vectors changes the sign of the answer.
Students also sometimes expect the result to lie in the same plane as the original vectors. It does not. The cross product is perpendicular to that plane.
Common checks for a cross product result:
The result must be a vector.
Its magnitude uses $\sin\theta$.
Its direction is perpendicular to both original vectors.
Changing the order changes the sign.
Summary
The cross product combines two vectors to make a new vector perpendicular to both. Its magnitude is
$$
|\vec{A} \times \vec{B}| = AB\sin\theta
$$
and its direction is given by the right hand rule. In component form,
$$
\vec{A} \times \vec{B}
=
\left(
A_y B_z - A_z B_y,\,
A_z B_x - A_x B_z,\,
A_x B_y - A_y B_x
\right)
$$
It is especially useful for describing areas, rotational effects, and certain forces in physics.