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Physics: High School18 chapters | 211 lessons

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Lesson Transcript

Instructor:
*David Wood*

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what a cross product is and calculate the cross product of two vectors. A short quiz will follow.

A **vector** is a quantity that has both magnitude (numerical size) and direction. Vectors are things like velocity, displacement, force, electric field. Those quantities always have a direction. A displacement isn't just 3 meters, it's 3 meters to the west. A **scalar** on the other hand (for example, temperature), only has a numerical value; it has no direction.

When we multiply two vectors together, the result can either be a vector or a scalar. Force multiplied by displacement equals work. And work is a scalar. Two vectors multiplied together give you a scalar. But a magnetic field multiplied by velocity (when also multiplied by charge) is equal to force. So in this equation, two vectors multiplied together give you a vector.

When the result of multiplying two vectors is a scalar, we've just completed a dot product. But if the result is a vector, then we have a cross product. A **cross product** is where you multiply one vector by the component of the second vector which acts at 90 degrees to the first vector.

So going back to the example from magnetism, the force on a moving charge inside an external magnetic field is proportional to the cross product between the magnetic field vector and the velocity vector for the charge. In other words, it's proportional to the magnetic field vector multiplied by the component of the velocity that acts at 90 degrees to the magnetic field vector. If the velocity is diagonal, moving at say, 30 degrees, you'll have to multiply the velocity by sine 30 to get the component of the velocity which acts at 90 degrees to the magnetic field. That's why the equation for this magnetic force has 'sine' in it.

The equation to calculate a cross product is pretty simple. The cross product between vectors *A* and *B* is equal to the magnitude of vector *A* multiplied by the magnitude of vector *B* multiplied by sine of the angle between them. So if you want the cross product of magnetic field and velocity, as I talked about earlier, you would take the magnitude of the magnetic field, multiply it by the magnitude of the velocity, and multiply that by sine of the angle between the magnetic field and velocity vectors. That will give you the magnitude of your answer. But your answer is, in itself, a vector. So what is the direction of your answer?

To get the direction, you have to use a right hand rule. I want you to give me a thumbs up with your right hand. When you do that, your fingers are curling in a particular direction. If you point your thumb towards the screen and look at the back of your fingers, they curl clockwise, for example.

Here is a diagram of the two vectors we're multiplying together:

We're multiplying vector *A* by vector *B*. To figure out the direction of your final answer, use the curl of your fingers to push (or wind) vector *A* towards vector *B*. When you do that, your thumb will point out of the page, and that direction is the direction of your final answer.

An important thing to notice here is that the order you write your cross product in doesn't impact the numerical answer, but it does impact the direction. *A*-cross-*B* gives you a direction out of the page. But if you wound your fingers the opposite way, for *B*-cross-*A*, you would have had a direction into the page. So unlike most multiplications, where the order you write the two things you're multiplying together doesn't matter, with cross products it does.

Maybe this would be easier using an example. Let's say we're trying to multiply a magnetic field vector, *v*, by a velocity vector, *B*. Vector *B* is pointing up, and vector *v* is pointing diagonally up and to the right, at an angle of 25 degrees to vector *B*. If the magnitude of vector *B* is 30 teslas, and the magnitude of vector *v* is 8 meters per second, what is the cross product of *v* and *B*?

First of all, let's write down what we know. *B* is equal to 30 teslas, and *v* is equal to 8 meters per second, and the angle between the vectors, theta, equals 25 degrees. So to find out the magnitude of the cross product, we just plug numbers into the equation and solve. 8 multiplied by 30 multiplied by sine 25 gives us a value of 101.4 tesla meters per second.

But what about the direction? Well, looking at the diagram, make a thumbs up, and use your fingers to find vector *v* onto vector *B*. If you do that, your thumb points out of the screen (or out of the page). So our final answer is 101.4 tesla meters per second out of the page.

And that's it; we're done.

A **vector** is a quantity that has both magnitude (numerical size) and direction. When we multiply two vectors together, the result can either be a vector or a scalar. When the result of multiplying two vectors is a scalar, that multiplication is a dot product. But if the result is a vector, then the multiplication is a cross product. A **cross product** is where you multiply one vector by the component of the second vector which acts at 90 degrees to the first vector.

The equation to calculate a cross product is pretty simple. The cross product between vectors *A* and *B* is equal to the magnitude of vector *A* multiplied by the magnitude of vector *B* multiplied by sine of the angle between them. That gives you the magnitude of your answer. But your answer is a vector in itself, so you also need to find the direction of your answer.

To get the direction, you have to use a right hand rule. If you make a thumbs-up with your right hand, you can use your fingers to push (or 'wind' or 'curl') vector *A* towards vector *B*. When you do that, your thumb will point in a particular direction, and that is the direction of your final answer. For the direction, *A*-cross-*B* won't give you the same result as *B*-cross-*A*, so the order you write your multiplication matters.

After studying this lesson, you will be able to:

- Define vector and cross product
- Calculate a cross product using the cross product equation
- Demonstrate how to get the direction of a cross product
- Recall why the order of multiplication matters for a cross product

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Physics: High School18 chapters | 211 lessons

- What Is a Vector? - Definition & Types 5:10
- Vector Addition (Geometric Approach): Explanation & Examples 4:32
- Resultants of Vectors: Definition & Calculation 6:35
- Scalar Multiplication of Vectors: Definition & Calculations 6:27
- Vector Subtraction (Geometric): Formula & Examples 5:48
- Standard Basis Vectors: Definition & Examples 5:48
- How to Do Vector Operations Using Components 6:29
- Vector Components: The Magnitude of a Vector 3:55
- Vector Components: The Direction of a Vector 3:34
- Vector Resolution: Definition & Practice Problems 5:36
- The Dot Product and Vectors: Definition & Formula 5:40
- Finding The Cross Product of Two Vectors 6:09
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