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Cross Product of Two Vectors

The cross product is a multiplication of two vectors in three-dimensional space. It basically combines two vectors, creating a third one, which will be perpendicular to both. Therefore, even if you consider a cross product of two vectors in 2D (with x and y components, for instance), the third vector will be in a different dimension (along the z-axis). You can learn more about the cross product and easily calculate different examples in our amazing cross product calculator 🇺🇸.

Along with this article, we are going to show you:

  • How to find the cross product of two vectors — properties and definitions;
  • How to calculate the cross product of two vectors — cross product rules;
  • How to do the cross product of two vectors using matrix notation;
  • The right-hand rule;
  • And more.

Before discussing the rules and formulas behind the cross product, let's consider some of its properties.

In vector algebra, there are two kinds of multiplication, the scalar product and the cross product. The scalar product can be thought of as a projection between two vectors, and the result is a scalar. On the other hand, the cross product of two vectors represents the projection of the area of a plane formed by these vectors in the perpendicular direction. Thus, the result of a cross product is a vector.

We know that these interpretations are not easy to follow when you start to study this subject; however, we will clarify them in this article.

Let us start by introducing the terminology. By writing the cross product a⃗×b⃗\vec{a} \times \vec{b}, we refer to it as "a cross b". As we pointed out, the result of the cross product will be a vector which we can name as c⃗\vec{c} . The absolute value of this vector is such that:

∣c⃗∣=∣a⃗∣ ∣b⃗∣sin⁡(θ)\left|\vec{c}\right| = |\vec{a}|\,|\vec{b}|\sin(\theta)

where θ\theta is the angle between a⃗\vec{a} and b⃗\vec{b}. This absolute value is equivalent to the area of a parallelogram generated by the vectors a⃗\vec{a} and b⃗\vec{b}, as we can see in the figure below. Now you can easily realize how the cross product is equivalent to the projection of this area in the perpendicular (or normal) direction to the plane formed by a⃗\vec{a} and b⃗\vec{b}.

Visualization of the cross product of 2 vectors

From the previous formula, we can verify that the cross product between two parallel vectors is zero and between two perpendicular vectors is maximum, which means, ∣c⃗∣=∣a⃗∣ ∣b⃗∣|\vec{c}| = |\vec{a}|\,|\vec{b}|.

🙋 You can learn more about other operations with vectors by checking our vector calculator 🇺🇸.

There are some rules that you can memorize to calculate the cross product of two vectors. In order to see these rules in action, let us consider the following vectors:

a⃗=axi^+ayj^+azk^b⃗=bxi^+byj^+bzk^\begin{split} \vec{a} &= a_x\hat{i}+a_y\hat{j}+a_z\hat{k} \\[1em] \vec{b} &= b_x\hat{i}+b_y\hat{j}+b_z\hat{k} \end{split}

where:

  • i^\hat{i} — Unit vector in the x direction;
  • j^\hat{j} — Unit vector in the y direction; and
  • k^\hat{k} — Unit vector in the z direction.

Therefore, the cross product between these vectors can be written as:

a⃗×b⃗=ax by(i^×j^)+ax bz(i^×k^)+ay bx(j^×i^)+ay bz(j^×k^)+az bx(k^×i^)+az by(k^×j^)\begin{split} \vec{a} \times \vec{b} &= a_x\,b_y(\hat{i}\times\hat{j}) + a_x\,b_z(\hat{i}\times\hat{k}) \\[1em] & +a_y\,b_x(\hat{j}\times\hat{i}) + a_y\,b_z(\hat{j}\times\hat{k})\\[1em] &+a_z\,b_x(\hat{k} \times \hat{i})+a_z\,b_y(\hat{k} \times \hat{j}) \end{split}

where we used the fact that the cross products of parallel unit vectors, such as (i^×i^)(\hat{i}\times\hat{i}), are equal to zero. The rules of the cross product are the following:

(i^×j^)=−(j^×i^)=k^(k^×i^)=−(i^×k^)=j^(j^×k^)=−(k^×j^)=i^\begin{split} & (\hat{i}\times\hat{j}) = - (\hat{j}\times\hat{i}) = \hat{k} \\[1em] & (\hat{k} \times \hat{i}) = - (\hat{i} \times \hat{k}) = \hat{j} \\[1em] & (\hat{j}\times\hat{k}) = - (\hat{k}\times\hat{j}) = \hat{i} \end{split}

Thus, by substituting the previous rules in the cross product, we obtain:

a⃗×b⃗=(ay bz−az by)i^+(az bx−ax bz)j^+(ax by−ay bx)k^\begin{split} \vec{a} \times \vec{b} &= \left(a_y\,b_z - a_z\,b_y \right)\hat{i} \\[1em] & + \left(a_z\,b_x - a_x\,b_z\right)\hat{j} \\[1em] &+\left(a_x\,b_y - a_y\,b_x\right)\hat{k} \end{split}

The same result of the cross product between a⃗×b⃗\vec{a} \times \vec{b} can be derived by computing the determinant of a matrix given by:

a⃗×b⃗=det⁡(i^j^k^axayazbxbybz)\vec{a} \times \vec{b} = \det\left(\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{matrix}\right)

This method is often used to find the resultant vector, since you do not need to memorize any rule. You can check an interesting application of the cross product in physics with the magnetic force on a current-carrying wire calculator 🇺🇸.

A useful way to predict the direction of the resultant vector, obtained from the cross product, is the famous right-hand rule. In order to apply the right-hand rule, you need to follow the steps below:

  • Align your index finger towards the direction of vector a⃗\vec{a};
  • Align your middle finger towards the direction of vector b⃗\vec{b}; and
  • Your thumb will point in the direction of the resultant vector a⃗×b⃗\vec{a}\times\vec{b}.

You can see all the steps in the figure below.

Right-hand rule, index finger pointing to a, middle finger pointing to b, and thumb pointing to the resultant vector.

There is another version of the right-hand rule that is broadly applied. You can simply put your right-hand fingers in the direction of the first vector, and then close a fist towards the direction of the second vector. As a result, your thumb is going to point in the direction of the resultant vector.

The cross product of two vectors a and b is a resultant vector, which is perpendicular to both vectors, or normal to the plane formed by a and b.

The final result of a cross product is a vector, while the final result of a dot product is a scalar. The dot product refers to the scalar product between vectors a and b, which is maximum if both vectors are parallel. On the other hand, the cross product between vectors a and b will have its maximum absolute value if both vectors are perpendicular to each other.

You can calculate the cross product of two 2D vectors following the steps below:

  1. Take vector a = (2, 4) in the (x, y) plane.
  2. Take vector b = (3, 5) in the (x, y) plane.
  3. Use the rules of the cross product to find that: c = (0, 0, -2), pointing in the z direction.

This article was written by João Rafael Lucio dos Santos and reviewed by Steven Wooding.