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Cross Product: Determinants

Previously, we introduced the cross product of two vectors in 3D space. In this section, we will explore the cross product in more detail and show how it can be computed using determinants.

Table of Contents​

Warmup: Cross Products in 2D​

In 2D space, the cross product of two vectors is not defined. However, we can make a modified definition of the cross product that does make sense in 2D space; one that just gives us the area of the parallelogram, but does not have a direction.

Then, this modified cross product of two vectors and in 2D space is simply the determinant of the matrix formed by the two vectors:

This gives us the area of the parallelogram spanned by the two vectors.

Cross Products in 3D​

As we discussed in the previous section, the cross product of two vectors and in 3D space is a vector that is orthogonal to both and .

Trying to compute the cross product of two vectors directly can be quite cumbersome. However, we can use the determinant to compute the cross product in a more elegant way.

Recall that the determinant, in 3D, gives the volume of the parallelepiped spanned by the three vectors. Given three vectors , , and , the determinant of the matrix can be written as:

Next, let's consider how this determinant can be computed geometrically:

Like a parallelogram, the volume of the parallelepiped won't change if you "tilt" it. This means that the volume of the parallelepiped is the same as the product of (1) the area of the base parallelogram, and (2) the vertical height of the parallelepiped:

The height is just the length of the projection of onto the normal, :

This is the best part - we can group and together to get a new vector, :

But wait... what is ? Its magnitude is the area of the parallelogram, and its direction is the normal to the parallelogram. This is exactly the cross product of and !

This means that the cross product of and is the vector that, when dotted with a third vector, gives the volume of the parallelepiped spanned by the three vectors. We already have another formula for this volume from the determinant, so we can equate the two:

This gives us the components of the cross product of and :

We can then write the cross product of and as:

Alternatively, going back to the determinant form, we can write the cross product as:

(Of course you can't place basis vectors in the elements of a matrix; this is just a notational trick to help us remember the formula.)

Finally, putting it all together, we have the formula for the cross product of two vectors in 3D space:

Summary and Next Steps​

In this section, we derived the formula for the cross product of two vectors in 3D space using determinants.

Here are the key points to remember:

  • In 2D, while the cross product is not defined, we can use the determinant of the matrix formed by the two vectors to get the area of the parallelogram.

  • In 3D, the cross product of two vectors can be obtained with a determinant, giving the vector that is orthogonal to both vectors:

In the next section, we will, just like with the dot product, explore some properties and corollaries of the cross product.