In this explainer, we will learn how to calculate the vector product of two vectors using both the components of the vectors and the magnitudes of the two vectors and the angle between them.

The **vector product** is an operation that can be applied to two vectors that
produce another vector.

The vector product is used in many different areas of physics. One calculation that it can be useful for is calculating the torque on an object.

Consider a car wheel that can freely rotate around its axis. A force is applied to the wheel at a point on the edge of the wheel. The vector from the center of the wheel to the point where the force acts is . The force acts along a tangent to the wheel. This is shown in the diagram below.

If the magnitude of the force is and the magnitude of the vector from the center of the wheel to the point where the force acts, which is just the radius, is , then the torque on the wheel, , is just equal to multiplied by ,

But what if the force does *not* act along a tangent to the wheel? The
diagram below shows the same scenario but with the force acting at an angle,
, to the tangent.

In this scenario, we cannot use to calculate the torque on the wheel. Instead, we can use the vector product of and to find the torque.

The vector product is notated with the times sign between the two vectors,

Because of this, the vector product is also called the **cross product**.

Note that since the result of the vector product is another vector, torque is a
vector quantity. When we use , as in the previous scenario, we are
only calculating the *magnitude* of the torque, but it does also have a
direction.

In order to understand how the vector product works, letβs first look at what the result of applying the vector product to unit vectors is.

Recall that a unit vector is a vector that has a magnitude of 1. We can represent any vector as a sum of unit vectors along the cardinal axes, which, when we are working in just two dimensions, are the - and -axes. Conventionally, is a unit vector that points along the -axis and is a unit vector that points along the -axis.

Working with the vector product, however, necessarily takes us into three dimensions, so we also need a unit vector that points along the -axis. Conventionally, this unit vector is represented by the symbol .

The diagram below shows the -, -, and -axes and their corresponding unit vectors: , , and .

Importantly, unlike with multiplication of simple numbers, with the vector product, the order of the two vectors in the product affects the result. So, for example, the result of is not equal to the result of :

The vector product is defined such that a vector product involving both and will point in the direction of or . The unit vectors and both lie in the -plane, and the result of the vector product is always normal to the plane formed by the two vectors that the vector product is being applied to.

In the case of , the result is just :

In the case of , the result is :

Notice that . Swapping the order of the unit vectors in the product gives a result with the same magnitude but points in the opposite direction. This is generally true for the vector product of any two vectors that the product is applied to.

### Rule: Swapping the Order of the Operands in the Vector Product

Swapping the order of the operands in the vector product gives a result that has the same magnitude but points in the opposite direction.

For any two vectors and ,

Furthermore, the vector product of either or with itself is equal to 0:

Knowing all of this, we can now work out a formula for the vector product of any two vectors that lie in the -plane. Consider a vector , which has components and , and a vector , which has components and :

The vector product of and is

As with ordinary multiplication, we can expand the brackets, giving us

At the moment, this is quite a long and complex expression. However, it is possible to simplify it drastically using the rules for the vector products of unit vectors we defined earlier. Recall that both and are equal to 0. This means that which simplifies to

Recall as well that and , so

We now have a simple formula for calculating the vector product. Note that this
formula will only work for two vectors that lie in the
-plane, which means that their -component
must be 0. The formula for calculating the vector product of *any*
two vectors is more complex; however, in this lesson, we will only be applying the
vector product to two vectors that lie in the -plane.

### Definition: The Vector Product of Two Vectors That Lie in the π₯π¦-Plane

If two vectors and both lie in the -plane, their vector product is given by where and are the components of and and are the components of .

Letβs now try using this formula in an example question.

### Example 1: Calculating the Vector Product of Two Vectors That Lie in the π₯π¦-plane given Their Components

Consider the two vectors and . Calculate .

### Answer

We can use the formula to work out the vector product of and . Substituting in the values, we get

The result of is . Note that both and lie in the -plane, while their vector product points along the -axis, normal to the -plane.

### Example 2: Calculating the Vector Product of Two Vectors That Lie in the π₯π¦-plane given Their Components

Consider the two vectors and .

- Calculate .
- Calculate .

### Answer

**Part 1**

We can use the formula to work out the vector product of and . Substituting in the values, we get

The result of is .

**Part 2**

In the first part of the question, we were asked to work out ; now, we are asked to work out the vector product of these two vectors but with the vectors going into the product the other way round.

The result of the product will not be the same as in the first part because with the vector product, the order affects the result. However, we do not need to work out the answer again numerically because we can just recall that, for any two vectors and ,

Reversing the order of the vectors produces the same result but multiplied by .

So, if , .

### Example 3: Calculating the Vector Product of Two Vectors Shown on a Grid

The diagram shows two vectors, and . Each of the grid squares in the diagram has a side length of 1. Calculate .

### Answer

In this question, we have been shown two vectors on a grid and asked to find their vector product.

We can work out the components of the vectors by looking at the grid. Vector has a horizontal length of 4 grid squares and a vertical length of 1 grid square. Therefore,

Vector has a horizontal length of 3 grid squares and a vertical length of 5 grid squares. Therefore,

We can then use the formula to work out the vector product of and . Substituting in the values, we get

The result of is .

So far, we have worked out the vector product of two vectors using algebraic manipulation of the vectorsβ components. However, there is also a way of defining the vector product geometrically.

### Definition: The Vector Product of Two Vectors

Consider two vectors and . The angle between the two vectors is . This is shown in the diagram below.

The vector product of and is equal to the magnitude of multiplied by the magnitude of , multiplied by the sine of the angle between them, , multiplied by a unit vector , which points normal to the plane formed by and ,

If we say that the magnitude of is and the magnitude of is , then we can write this as

It may not look like it at first glance, but this formula does actually produce the same result as the formula for the vector product that we used earlier. The equivalence of these two formulae can be proven algebraically, but this is beyond what we are going to do in this lesson.

Seeing the vector product defined in this way tells us some useful properties of the vector product.

Firstly, recall the shape of a sine curve. This is shown on the graph below.

When , . This means that when the angle between two vectors is , which means that they are parallel, their vector product is 0.

Similarly, when , . This means that when the angle between two vectors is , which means that they are antiparallel, their vector product is 0.

### Rule: The Vector Product of Parallel or Antiparallel Vectors

If two vectors point in the same direction or the opposite direction, their vector product is zero.

This further means that the vector product of any vector with itself is 0, since any vector is parallel to itself.

### Rule: The Vector Product of Any Vector with Itself

The vector product of any vector with itself is zero:

The sine function has its maximum value of 1 when . This means that the vector product of two vectors will have its largest value when the two vectors are at right angles to each other. This is the opposite of the scalar product, which has a value of 0 when the two vectors are at right angles to each other.

Consequently, we can think of the vector product as being a measure of two things: how large the vectors are and the extent to which they are at right angles to each other. The greater the magnitude of either vector, the greater the magnitude of their vector product. The closer the angle between them, , is to , the greater the magnitude of the vector product.

But how can we work out the direction of ? Consider again two vectors and , as shown in the diagram below.

Firstly, it is important to remember that when we talk about the βangle between two vectors,β we are referring to the smaller of the two angles formed by the two vector arrows. This term does not refer to the larger angle, labeled in the diagram below.

In the diagram above, is
*counterclockwise* of . In other words,
if we had to rotate vector by
, the angle between the vectors, so that it pointed in the
same direction as , we would have to rotate it
*counterclockwise*. In this case, for
,
points **out of the screen**, along the
positive -axis, and is equal to
.

In the diagram below, the reverse is true and is
*clockwise* of . In other words, if we had
to rotate vector by so
that it pointed in the same direction as ,
we would have to rotate it *clockwise*. In this case, for
,
points **into the page**, along the
negative -axis, and is equal to
.

### Rule: The Direction of the Vector Product for Vectors in the π₯π¦-Plane

For two vectors in the -plane, the direction of is if is counterclockwise of and if is clockwise of .

### Example 4: Calculating the Vector Product of Two Vectors given Their Magnitudes and the Angle between Them

The diagram shows two vectors, and . Calculate the magnitude of the vector product of and . Give your answer to the nearest integer.

### Answer

We can use the formula where is the magnitude of and is the magnitude of , to find the vector product of and .

In this case, we are asked to find only the magnitude of the vector product. Taking the magnitude of both sides of the above formula gives us

The vector has no effect on the magnitude of the vector product, as it is a unit vector and so itself has a magnitude of 1.

Substituting in the values given in the question, we get

Rounded to the nearest integer, this is 190.

### Example 5: Finding the Vector Product of Two Vectors Shown on a 3D Grid

The diagram shows two vectors, and , in three-dimensional space. Both vectors lie in the -plane. Each of the squares of the grid has a side length of 1. Calculate .

### Answer

This question shows us a 3D space but the two vectors lie in the -plane.

The key to answering this question is in noticing that the vectors are antiparallel to each other. From the diagram, we can see that has components and has components ; therefore, .

Recall that, for two vectors that are parallel or antiparallel, their vector product is 0, so the answer is 0.

### Key Points

- The vector product is an operation that can be applied to two vectors that produce another vector.
- The vector product is also called the cross product.
- The result of is .
- The result of is .
- For two vectors in the -plane, if we know the components of the vectors, we can calculate their vector product using
- If we know the magnitudes of the two vectors and the angle between them, we can calculate their vector product using
- If two vectors are parallel or antiparallel, their vector product is 0.
- The vector product has its maximum magnitude when two vectors are at right angles to each other.