How to Calculate the Cross Product in Python

In this tutorial, you’ll learn how to use Python to calculate the cross-product. In particular, you’ll learn how to calculate the cross-product using the popular library, NumPy, and how to calculate it from scratch. The cross product is a common mathematical vector operation that takes two vectors as input and produces a third vector that is perpendicular to both of them.

By the end of this tutorial, you’ll have learned how to do the following:

• What the motivation behind the cross-product is
• How to calculate the cross-product using NumPy
• How to develop a custom function to calculate the cross-product in Python

Want to learn about the dot product in Python instead? Check out my post on it!

Calculating the Cross Product Using NumPy

The simplest way to calculate the cross-product in Python is to use the NumPy library. NumPy allows you to easily compute linear algebra calculations, including the cross-product. In order to do this, NumPy provides the np.cross() function.

Let’s take a look at how the NumPy cross function works:

# Understanding the NumPy cross Function
numpy.cross(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None)

We can see that the function has two required parameters, a and b. These represent the two vectors that you want to calculate the cross-product for. Let’s now take a look at an example of how to use the function.

Example of Calculating the Cross Product Using NumPy

The only required parameters for the np.cross() function are the two arrays for which you want to calculate the cross-product. Let’s create some NumPy arrays and pass them into the np.cross() function.

# Calculating the Cross Product in NumPy
import numpy as np
a = np.array([3,2,1])
b = np.array([6,5,4])

cross_prod = np.cross(a, b)
print(cross_prod)

# Returns:
# [ 3 -6  3]

In the example above, we first declared two arrays, a and b. We then passed these into the cross function to calculate the cross product. Let’s now dive into how we can better understand the cross-product itself.

Understanding the Cross Product

Before diving into calculating the cross-product in Python using a custom function, let’s quickly explore what the cross-product is and how it’s often used.

A vector is a quantity that has both magnitude and direction. Vectors are often represented graphically as arrows, where the length of the arrow represents the magnitude of the vector. Similarly, the direction of the arrow represents the direction of the vector.

The cross-product is an operation that takes two vectors as input and produces a third vector that is perpendicular to both of them. The resulting vector is also known as the vector product or the outer product.

The formula for the cross-product can be displayed written as shown below:

[A1, A2, A3] x [B1, B2, B3] = [A2*B3 - A3*B2, A3*B1 - A1*B3, A1*B2 - A2*B1]

Some important properties of the cross-product include:

• The cross-product is not commutative, which means that u x v is not equal to v x u.
• The cross product is distributive, which means that u x (v + w) = u x v + u x w.
• The cross product is anticommutative, which means that u x v = -v x u.

Calculating the Cross Product Using a Custom Python Function

Now, let’s take a look at how we can create a custom function. While I’m a big advocate for using NumPy, since it’s well-tested and much faster, it can be a good idea to understand how to calculate the cross-product using a custom function.

In my experience, this approach is good primarily for interviews. Let’s take a look at what this function looks like:

# Defining a Custom Function 
def cross_product(vector1, vector2):
    """
    Calculates the cross product of two vectors.

    Args:
        vector1 (list): A list of three floats representing the first vector.
        vector2 (list): A list of three floats representing the second vector.

    Returns:
        A list of three floats representing the cross product of the two vectors.
    """
    x = vector1[1] * vector2[2] - vector1[2] * vector2[1]
    y = vector1[2] * vector2[0] - vector1[0] * vector2[2]
    z = vector1[0] * vector2[1] - vector1[1] * vector2[0]
    return [x, y, z]

In the function itself, we calculate the cross-product by accessing the indices of the different lists, multiplying them, and subtracting them. You could improve the function by adding some assertions and length checks. Let’s take a look at an example of using this function:

Example of Calculating the Cross Product Using a Custom Python Function

In this section, we’ll take a look at how to use the custom function we just built. We’ll use the same values that we used in the NumPy example. This will allow us to see if the function works as expected.

In the code block below, we pass in two lists: [3,2,1] and [6,5,4]. Let’s see what this looks like:

# Calculating a Cross Product with a Custom Function
print(cross_product([3,2,1], [6,5,4]))

# Returns: [3, -6, 3]

We can see that by passing in our two lists we return the same cross-product as we did before.

Frequently Asked Questions

Conclusion

In conclusion, we have learned how to calculate the cross-product using Python, both with NumPy and a custom function. The cross-product is a critical mathematical operation that allows us to calculate the area of a parallelogram, the direction of a magnetic field, and the torque on an object, among other things. By understanding the cross-product and its applications, we can solve a wide range of problems in science, engineering, and computer graphics. Whether you choose to use NumPy or a custom function, Python provides a powerful and flexible tool for calculating the cross-product.

Check out the official documentation for the NumPy cross function.