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Finding The Cross Product of Two Vectors

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  • 0:00 What Is a Cross Product?
  • 1:49 Equation for Cross Product
  • 3:32 Example Calculations
  • 4:48 Lesson Summary
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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.

What is a Cross Product?

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.

Two vectors multiplied can equal a vector or a scalar
Vector and Scalar Equations

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.

Calculating magnetic force
Cross Product Example

Equation for Cross Product

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?

Equation for a Cross Product
The Equation for a Cross Product

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:

Vector diagram
vector diagram for example

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.

Use the right hand rule for direction
Direction of Vector

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.

The order of multiplication affects direction
Multiplication affects direction

Example Calculations

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?

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