Cross Product & Right Hand Rule: Definition, Formula & Examples Video

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  • 0:00 What Is A Vector?
  • 1:09 Vector Multiplication
  • 2:25 Applications Of The…
  • 4:00 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Multiplying two vectors will sometimes give you another vector, known as a cross product, which has many important applications in the real world. In this lesson, you'll learn how to use the right hand rule to find the cross product while exploring some examples.

What Is a Vector?

The word vector has many different meanings in English, but in math and physics, a vector refers to something very specific. In these fields, a vector is a mathematical representation of a physical quantity that has both a magnitude and a direction, as indicated by arrows. It can also be broken down into components that show how much of the vector exists in each direction.

A vector can have components in three dimensions. Sub x, sub y and sub z represent the x, y and z components of vector a.
3D Vectors

As you can see, a vector can have components in three dimensions: sub x, sub y and sub z represent the x, y and z components of vector a. So, what kinds of quantities can be represented by vectors? While there are many, the easiest to understand is most likely position. A position vector is a vector that tells you where an object is located relative to some origin.

For example, let's say John's house is 5 miles from Anna's. However, this doesn't tell Anna exactly where his house is located. To actually find his house, Anna needs to know both its magnitude, or how far away it is, and its direction, in this case, 40 degrees north of east.

Vector Multiplication

There are two ways to multiply vectors, which lead to the cross product. A cross product tells you what part of one vector is perpendicular to the other vector.

The cross product of vectors a and b is another vector perpendicular to both a and b.
Cross product diagram

Here, the cross product of vectors a and b is another vector perpendicular to both a and b. If you know the magnitude of vector a and vector b, you can find the magnitude of the cross product by multiplying the magnitude of a, the magnitude of b and the sine of the angle between them.

cross product definition

This equation will give you the magnitude of the cross product vector, but wait! Remember that vectors always have a magnitude and a direction. How do we find the direction? For that, we need the right hand rule.

To use the right hand rule, you first have to hold up your right hand. Make sure it's not your left, or it won't work! Hold your index finger, middle finger and thumb so that they are all perpendicular to each other, like an x, y and z coordinate system. Now, rotate your hand so that your index finger points in the direction of vector a and your middle finger points in the direction of vector b. Your thumb will point in the direction of the cross product a x b.

Right Hand Rule
right hand rule

Be careful when calculating a cross product because it's easy to mix up the vectors. However, the order is important: a x b is not equal to b x a.

Applications of the Cross Product

One important application of the cross product is in the calculation of torque. Torque is the tendency of a force to cause an object to rotate. When you push on the end of a wrench, you are applying torque to make it rotate. Only the part of the force perpendicular to the wrench will create torque, and remember, that's exactly what the cross product tells you!

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