There are two main formulas that can be applied to find the cross product of two vectors. The first formula uses the magnitudes of the vectors and the angle between their directions. The second formula uses the components of the vectors.

Here, we will learn the two formulas for calculating the cross product of two vectors. Then, we will look at some of the most important properties of the cross product.

##### PHYSICS

**Relevant for**…

Learning about the formulas and properties of the cross product of vectors.

##### PHYSICS

**Relevant for**…

Learning about the formulas and properties of the cross product of vectors.

## Cross product using magnitudes and angle between vectors

The cross product of two vectors $latex \vec{A}$ and $latex \vec{B}$ is denoted by $latex \vec{A} \times \vec{B}$. The result of the cross product is a vector.

When we have the magnitudes of the vectors and the angle between their directions, the magnitude of their cross product is calculated with the following formula:

$$\vec{A}\times \vec{B}=AB\sin(\theta)$$

where $latex A$ and $latex B$ are the magnitudes of $latex \vec{A}$ and $latex \vec{B}$ respectively, and $latex \theta$ is the angle between the vectors.

Let’s use the following diagram to give a geometric interpretation of the cross product:

We see that $latex B \sin(\theta)$ is the component of $latex \vec{B}$ that is perpendicular to the direction of $latex \vec{A}$.

Alternatively, the magnitude of $latex \vec{A}\times \vec{B}$ is the magnitude of $latex \vec{B}$ multiplied by the component of $latex \vec{A}$ perpendicular to the direction of $latex \vec{B}$.

### Direction of the cross product

The result of the cross product is a vector, so it has a magnitude and a direction.

The direction of the cross product is perpendicular to the plane containing the vectors $latex \vec{A}$ and $latex \vec{B}$. That is, the direction of the cross product is perpendicular to both vectors.

However, there are always two directions perpendicular to a given plane, one on each side of the plane. We can choose the correct direction as follows:

We use our right hand to rotate from the vector $latex \vec{A}$ to the direction of the vector $latex \vec{B}$. Our fingers should point in the direction of rotation, as shown in the figure:

The direction of $latex \vec{A}\times \vec{B}$ corresponds to the direction in which our thumb is pointing.

Similarly, we can find the direction of $latex \vec{B}\times \vec{A}$ by rotating from vector $latex \vec{A}$ to the direction of vector $latex \vec{B}$, as the picture shows:

We see that $latex \vec{B}\times \vec{A}$ is the opposite of $latex \vec{A}\times \vec{B}$. That is, we have:

$latex \vec{A}\times \vec{B}=-\vec{B}\times \vec{A}$

## Cross product of vectors using their components

The components of the cross product of two vectors $latex \vec{A} \cdot \vec{B}$ can be calculated with the following formula if we know the $latex x,~y,~z$ components of the vectors:

$latex C_{x}=A_{y}B_{z}-A_{z}B_{y}$

$latex C_{y}=A_{z}B_{x}-A_{x}B_{z}$

$latex C_{z}=A_{x}B_{y}-A_{y}B_{x}$

To prove this formula, we need to find the cross product of the unit vectors $latex \hat{i},~\hat{j},~\hat{k}$.

The cross product of any vector by itself is 0, so:

$latex \hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=0$

Now, we use the right-hand rule to find:

$latex \hat{i}\cdot \hat{j}=-\hat{j}\cdot \hat{i}=\hat{k}$

$latex \hat{j}\cdot \hat{k}=-\hat{k}\cdot \hat{j}=\hat{i}$

$latex \hat{k}\cdot \hat{i}=-\hat{i}\cdot \hat{k}=\hat{j}$

Now, we write the vectors in terms of their components, expand the product and apply the results found above to simplify:

$$\vec{A}\times \vec{B}=(A_{x}\hat{i}+A_{y}\hat{j}+A_{z}\hat{k})\times (B_{x}\hat{i}+B_{y}\hat{j}+B_{z}\hat{k})$$

$$=A_{x}\hat{i}\times B_{x}\hat{i}+A_{x}\hat{i}\times B_{y}\hat{j}+A_{x}\hat{i}\times B_{z}\hat{k})$$

$$+A_{y}\hat{j}\times B_{x}\hat{i}+A_{y}\hat{j}\times B_{y}\hat{j}+A_{y}\hat{j}\times B_{z}\hat{k})$$

$$+A_{z}\hat{k}\times B_{x}\hat{i}+A_{z}\hat{k}\times B_{y}\hat{j}+A_{z}\hat{k}\times B_{z}\hat{k})$$

Here, we can rewrite the individual terms as $latex A_{x}\hat{i}\times B_{y}\hat{j}=A_{x}B_{y}\hat{i}\times\hat{ j}$ and so on. By regrouping and simplifying, we have:

$latex \vec{A}\times \vec{B}=(A_{y}B_{z}-A_{z}B_{y})\hat{i}+(A_{z}B_{x}-A_{x}B_{z})\hat{j}+(A_{x}B_{y}-A_{y}B_{x})\hat{k}$

## Properties of the cross product of vectors

The most important properties of the cross product of vectors are the following:

### Orthogonality

The result of the cross product is a vector orthogonal (perpendicular) to the two initial vectors. If $latex \vec{C}= \vec{A} \times \vec{B}$, then the dot product of $latex \vec{C}$ with $latex \vec{A}$ or $latex \vec {B}$ will be zero, that is,

$latex \vec{C}\cdot \vec{A} = 0$

and

$latex \vec{C}\cdot \vec{B} = 0$

### Non-commutativity

The cross product is not commutative, which means that the order of the vectors does matter. For two vectors $latex \vec{A}$ and $latex \vec{B}$, $latex \vec{A} \times \vec{B}$ is not equal to $latex \vec{B} \times \ vec{A}$. In fact,

$latex \vec{A} \times \vec{B} = – \vec{B} \times \vec{A}$

### Distributivity

The cross product is distributive over the sum of the vectors. That is, we have

$latex \vec{A} \times (\vec{B} + \vec{C}) = (\vec{A} \times \vec{B}) + (\vec{A} \times \vec{C})$

### Scalar multiplication

The cross product is compatible with scalar multiplication. If $latex \vec{A}$ and $latex \vec{B}$ are vectors and $latex k$ is a scalar, then

$latex k(\vec{A} \times \vec{B}) = (k\vec{A}) \times \vec{B} = \vec{A} \times (k\vec{B})$

### Parallel vectors

If two vectors $latex \vec{A}$ and $latex \vec{B}$ are parallel or antiparallel, their cross product will be the zero vector (0, 0, 0).

### Cross product of a vector by itself

From the previous property, it follows that the product of a vector by itself is equal to 0:

$latex \vec{A}\times \vec{A}=0$

### Right-hand rule

The direction of the cross product can be determined using the right-hand rule. If you point the index finger of the right hand in the direction of the first vector ($latex \vec{A}$) and the middle finger in the direction of the second vector ($latex \vec{B}$), the thumb will point in the direction of the cross product ($latex \vec{A} \times \vec{B}$).

### Eriple scalar product

Given three vectors $latex \vec{A},~ \vec{B}~$ and $latex ~\vec{C}$, the triple scalar product is defined as the cross product of $latex \vec{A}$ with the dot product of $latex \vec{B}~$ and $latex ~\vec{C}$, that is, $latex \vec{A} \times \vec{B} \cdot \vec{C}$.

This quantity is equal to the volume of the parallelepiped formed by the three vectors and has the property that

$latex \vec{A} \times \vec{B} \cdot \vec{C}=\vec{B} \times \vec{C} \cdot \vec{A}=\vec{C} \times \vec{A} \cdot \vec{B}$

### Collinearity test

If $latex \vec{A} \times \vec{B} = 0$ and at least one of the vectors is nonzero, then $latex \vec{A}~$ and $latex ~\vec{B}$ They are collinear, that is, they lie on the same line in the plane.

## See also

Interested in learning more about vectors? You can visit the following pages.

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