Lesson Explainer: The Vector Product of Two Vectors | Nagwa Lesson Explainer: The Vector Product of Two Vectors | Nagwa

Lesson Explainer: The Vector Product of Two Vectors Physics • First Year of Secondary School

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 ⃑𝐹 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 βƒ‘π‘Ÿ. The force acts along a tangent to the wheel. This is shown in the diagram below.

If the magnitude of the force is 𝐹 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 π‘Ÿ, then the torque on the wheel, 𝜏, is just equal to 𝐹 multiplied by π‘Ÿ, 𝜏=πΉπ‘Ÿ.

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, πœƒ, to the tangent.

In this scenario, we cannot use 𝜏=πΉπ‘Ÿ to calculate the torque on the wheel. Instead, we can use the vector product of ⃑𝐹 and βƒ‘π‘Ÿ to find the torque.

The vector product is notated with the times sign between the two vectors, 𝜏=βƒ‘π‘ŸΓ—βƒ‘πΉ.

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 𝜏=πΉπ‘Ÿ, 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, let’s 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 π‘₯- and 𝑦-axes. Conventionally, ⃑𝑖 is a unit vector that points along the π‘₯-axis and ⃑𝑗 is a unit vector that points along the 𝑦-axis.

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

The diagram below shows the π‘₯-, 𝑦-, and 𝑧-axes and their corresponding unit vectors: ⃑𝑖, ⃑𝑗, and βƒ‘π‘˜.

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 ⃑𝑖×⃑𝑗 is not equal to the result of ⃑𝑗×⃑𝑖: ⃑𝑖×⃑𝑗≠⃑𝑗×⃑𝑖.

The vector product is defined such that a vector product involving both ⃑𝑖 and ⃑𝑗 will point in the direction of βƒ‘π‘˜ or βˆ’βƒ‘π‘˜. The unit vectors ⃑𝑖 and ⃑𝑗 both lie in the π‘₯𝑦-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 ⃑𝑖×⃑𝑗, the result is just βƒ‘π‘˜: ⃑𝑖×⃑𝑗=βƒ‘π‘˜.

In the case of ⃑𝑗×⃑𝑖, the result is βˆ’βƒ‘π‘˜: ⃑𝑗×⃑𝑖=βˆ’βƒ‘π‘˜.

Notice that ⃑𝑗×⃑𝑖=βˆ’βƒ‘π‘–Γ—βƒ‘π‘—. 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 ⃑𝐴 and ⃑𝐡, ⃑𝐴×⃑𝐡=βˆ’βƒ‘π΅Γ—βƒ‘π΄.

Furthermore, the vector product of either ⃑𝑖 or ⃑𝑗 with itself is equal to 0: ⃑𝑖×⃑𝑖=0,⃑𝑗×⃑𝑗=0.

Knowing all of this, we can now work out a formula for the vector product of any two vectors that lie in the π‘₯𝑦-plane. Consider a vector ⃑𝐴, which has components 𝐴 and 𝐴, and a vector ⃑𝐡, which has components 𝐡 and 𝐡: ⃑𝐴=𝐴⃑𝑖+𝐴⃑𝑗,⃑𝐡=𝐡⃑𝑖+𝐡⃑𝑗.ο—ο˜ο—ο˜

The vector product of ⃑𝐴 and ⃑𝐡 is ⃑𝐴×⃑𝐡=𝐴⃑𝑖+𝐴⃑𝑗×𝐡⃑𝑖+𝐡⃑𝑗.ο—ο˜ο—ο˜

As with ordinary multiplication, we can expand the brackets, giving us ⃑𝐴×⃑𝐡=𝐴𝐡⃑𝑖×⃑𝑖+𝐴𝐡⃑𝑖×⃑𝑗+𝐴𝐡⃑𝑗×⃑𝑖+𝐴𝐡⃑𝑗×⃑𝑗.ο—ο—ο—ο˜ο˜ο—ο˜ο˜

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 ⃑𝑖×⃑𝑖 and ⃑𝑗×⃑𝑗 are equal to 0. This means that ⃑𝐴×⃑𝐡=0𝐴𝐡+𝐴𝐡⃑𝑖×⃑𝑗+𝐴𝐡⃑𝑗×⃑𝑖+0𝐴𝐡,ο—ο—ο—ο˜ο˜ο—ο˜ο˜ which simplifies to ⃑𝐴×⃑𝐡=𝐴𝐡⃑𝑖×⃑𝑗+𝐴𝐡⃑𝑗×⃑𝑖.ο—ο˜ο˜ο—

Recall as well that ⃑𝑖×⃑𝑗=βƒ‘π‘˜ and ⃑𝑗×⃑𝑖=βˆ’βƒ‘π‘˜, so ⃑𝐴×⃑𝐡=π΄π΅βƒ‘π‘˜+π΄π΅ο€»βˆ’βƒ‘π‘˜ο‡βƒ‘π΄Γ—βƒ‘π΅=π΄π΅βƒ‘π‘˜βˆ’π΄π΅βƒ‘π‘˜βƒ‘π΄Γ—βƒ‘π΅=ο€Ήπ΄π΅βˆ’π΄π΅ο…βƒ‘π‘˜.ο—ο˜ο˜ο—ο—ο˜ο˜ο—ο—ο˜ο˜ο—

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 π‘₯𝑦-plane, which means that their 𝑧-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 π‘₯𝑦-plane.

Definition: The Vector Product of Two Vectors That Lie in the π‘₯𝑦-Plane

If two vectors ⃑𝐴 and ⃑𝐡 both lie in the π‘₯𝑦-plane, their vector product is given by ⃑𝐴×⃑𝐡=ο€Ήπ΄π΅βˆ’π΄π΅ο…βƒ‘π‘˜,ο—ο˜ο˜ο— where 𝐴 and 𝐴 are the components of ⃑𝐴 and 𝐡 and 𝐡 are the components of ⃑𝐡.

Let’s now try using this formula in an example question.

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

Consider the two vectors ⃑𝑅=3⃑𝑖+2⃑𝑗 and ⃑𝑆=5⃑𝑖+8⃑𝑗. Calculate ⃑𝑅×⃑𝑆.

Answer

We can use the formula ⃑𝑅×⃑𝑆=ο€Ήπ‘…π‘†βˆ’π‘…π‘†ο…βƒ‘π‘˜ο—ο˜ο˜ο— to work out the vector product of ⃑𝑅 and ⃑𝑆. Substituting in the values, we get ⃑𝑅×⃑𝑆=(3Γ—8βˆ’2Γ—5)βƒ‘π‘˜βƒ‘π‘…Γ—βƒ‘π‘†=(24βˆ’10)βƒ‘π‘˜βƒ‘π‘…Γ—βƒ‘π‘†=14βƒ‘π‘˜.

The result of ⃑𝑅×⃑𝑆 is 14βƒ‘π‘˜. Note that both ⃑𝑅 and ⃑𝑆 lie in the π‘₯𝑦-plane, while their vector product points along the 𝑧-axis, normal to the π‘₯𝑦-plane.

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

Consider the two vectors ⃑𝐢=15⃑𝑖+7⃑𝑗 and ⃑𝐷=4⃑𝑖+9⃑𝑗.

  1. Calculate ⃑𝐢×⃑𝐷.
  2. Calculate ⃑𝐷×⃑𝐢.

Answer

Part 1

We can use the formula ⃑𝐢×⃑𝐷=ο€ΉπΆπ·βˆ’πΆπ·ο…βƒ‘π‘˜ο—ο˜ο˜ο— to work out the vector product of ⃑𝐢 and ⃑𝐷. Substituting in the values, we get ⃑𝐢×⃑𝐷=(15Γ—9βˆ’7Γ—4)βƒ‘π‘˜βƒ‘πΆΓ—βƒ‘π·=(135βˆ’28)βƒ‘π‘˜βƒ‘πΆΓ—βƒ‘π·=107βƒ‘π‘˜.

The result of ⃑𝐢×⃑𝐷 is 107βƒ‘π‘˜.

Part 2

In the first part of the question, we were asked to work out ⃑𝐢×⃑𝐷; 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 ⃑𝐴 and ⃑𝐡, ⃑𝐴×⃑𝐡=βˆ’βƒ‘π΅Γ—βƒ‘π΄.

Reversing the order of the vectors produces the same result but multiplied by βˆ’1.

So, if ⃑𝐢×⃑𝐷=107βƒ‘π‘˜, ⃑𝐷×⃑𝐢=βˆ’107βƒ‘π‘˜.

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

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

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 ⃑𝐴 has a horizontal length of 4 grid squares and a vertical length of 1 grid square. Therefore, ⃑𝐴=4⃑𝑖+1⃑𝑗.

Vector ⃑𝐡 has a horizontal length of 3 grid squares and a vertical length of 5 grid squares. Therefore, ⃑𝐡=3⃑𝑖+5⃑𝑗.

We can then use the formula ⃑𝐴×⃑𝐡=ο€Ήπ΄π΅βˆ’π΄π΅ο…βƒ‘π‘˜ο—ο˜ο˜ο— to work out the vector product of ⃑𝐴 and ⃑𝐡. Substituting in the values, we get ⃑𝐴×⃑𝐡=(4Γ—5βˆ’1Γ—3)βƒ‘π‘˜βƒ‘π΄Γ—βƒ‘π΅=(20βˆ’3)βƒ‘π‘˜βƒ‘π΄Γ—βƒ‘π΅=17βƒ‘π‘˜.

The result of ⃑𝐴×⃑𝐡 is 17βƒ‘π‘˜.

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

Definition: The Vector Product of Two Vectors

Consider two vectors ⃑𝐴 and ⃑𝐡. The angle between the two vectors is πœƒ. This is shown in the diagram below.

The vector product of ⃑𝐴 and ⃑𝐡 is equal to the magnitude of ⃑𝐴 multiplied by the magnitude of ⃑𝐡, multiplied by the sine of the angle between them, πœƒ, multiplied by a unit vector ⃑𝑛, which points normal to the plane formed by ⃑𝐴 and ⃑𝐡, ⃑𝐴×⃑𝐡=‖‖⃑𝐴‖‖‖‖⃑𝐡‖‖(πœƒ)⃑𝑛.sin

If we say that the magnitude of ⃑𝐴 is 𝐴 and the magnitude of ⃑𝐡 is 𝐡, then we can write this as ⃑𝐴×⃑𝐡=𝐴𝐡(πœƒ)⃑𝑛.sin

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 πœƒ=0∘, sin(πœƒ)=0. This means that when the angle between two vectors is 0∘, which means that they are parallel, their vector product is 0.

Similarly, when πœƒ=180∘, sin(πœƒ)=0. This means that when the angle between two vectors is 180∘, 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 ⃑𝐴 with itself is zero: ⃑𝐴×⃑𝐴=0.

The sine function has its maximum value of 1 when πœƒ=90∘. 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, πœƒ, is to 90∘, the greater the magnitude of the vector product.

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

Firstly, it is important to remember that when we talk about the β€œangle between two vectors,” 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 πœ™ in the diagram below.

In the diagram above, ⃑𝐡 is counterclockwise of ⃑𝐴. In other words, if we had to rotate vector ⃑𝐴 by πœƒ, the angle between the vectors, so that it pointed in the same direction as ⃑𝐡, we would have to rotate it counterclockwise. In this case, for ⃑𝐴×⃑𝐡, ⃑𝑛 points out of the screen, along the positive 𝑧-axis, and is equal to βƒ‘π‘˜.

In the diagram below, the reverse is true and ⃑𝐡 is clockwise of ⃑𝐴. In other words, if we had to rotate vector ⃑𝐴 by πœƒ so that it pointed in the same direction as ⃑𝐡, we would have to rotate it clockwise. In this case, for ⃑𝐴×⃑𝐡, ⃑𝑛 points into the page, along the negative 𝑧-axis, and is equal to βˆ’βƒ‘π‘˜.

Rule: The Direction of the Vector Product for Vectors in the π‘₯𝑦-Plane

For two vectors in the π‘₯𝑦-plane, the direction of ⃑𝐴×⃑𝐡 is βƒ‘π‘˜ if ⃑𝐡 is counterclockwise of ⃑𝐴 and βˆ’βƒ‘π‘˜ if ⃑𝐡 is clockwise of ⃑𝐴.

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

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

Answer

We can use the formula ⃑𝐴×⃑𝐡=𝐴𝐡(πœƒ)⃑𝑛,sin where 𝐴 is the magnitude of ⃑𝐴 and 𝐡 is the magnitude of ⃑𝐡, to find the vector product of ⃑𝐴 and ⃑𝐡.

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 ‖‖⃑𝐴×⃑𝐡‖‖=||𝐴𝐡(πœƒ)⃑𝑛||‖‖⃑𝐴×⃑𝐡‖‖=𝐴𝐡(πœƒ).sinsin

The vector ⃑𝑛 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 ‖‖⃑𝐴×⃑𝐡‖‖=12Γ—16Γ—(82)‖‖⃑𝐴×⃑𝐡‖‖=190.131….sin∘

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, ⃑𝐢 and ⃑𝐷, in three-dimensional space. Both vectors lie in the π‘₯𝑦-plane. Each of the squares of the grid has a side length of 1. Calculate ⃑𝐢×⃑𝐷.

Answer

This question shows us a 3D space but the two vectors lie in the π‘₯𝑦-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 ⃑𝐢 has components 4⃑𝑖+5⃑𝑗 and ⃑𝐷 has components βˆ’4βƒ‘π‘–βˆ’5⃑𝑗; therefore, ⃑𝐷=βˆ’βƒ‘πΆ.

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 ⃑𝑖×⃑𝑗 is βƒ‘π‘˜.
  • The result of ⃑𝑗×⃑𝑖 is βˆ’βƒ‘π‘˜.
  • For two vectors in the π‘₯𝑦-plane, if we know the components of the vectors, we can calculate their vector product using ⃑𝐴×⃑𝐡=ο€Ήπ΄π΅βˆ’π΄π΅ο…βƒ‘π‘˜.ο—ο˜ο˜ο—
  • If we know the magnitudes of the two vectors and the angle between them, we can calculate their vector product using ⃑𝐴×⃑𝐡=𝐴𝐡(πœƒ)⃑𝑛.sin
  • 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.

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