Vectors: Definition, Types & Examples

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  • 0:05 What Are Vectors?
  • 0:54 Position Vectors
  • 1:11 Magnitude of a Vector
  • 1:52 Unit Vectors
  • 2:27 Equal vs. Parallel Vectors
  • 4:05 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

In this lesson, learn what vectors are and how to represent them mathematically. You will also learn how to tell if two vectors are equal, parallel, collinear, or coplanar.

What Are Vectors?

What if I told you to go out of your house and walk for exactly one mile? You would probably quickly ask me, ''A mile in which direction?'' Of course, to really know where you'll end up after walking, you need to know two things: the distance you need to walk and the direction you need to go.

This is an example of a vector, which is a quantity with both a magnitude and a direction. There are many physical quantities that are vectors, including velocity, force, momentum, and electric field, so it is important to know how to recognize and use vectors.

You can represent a vector graphically with an arrow, like this one that you can see drawn below:

examples of vectors

As you can see here, vectors are designated by putting an arrow over the symbol that represents the vector. That lets you know that this quantity is a vector.

Position Vectors

One important type of vector is a position vector that gives the position of an object relative to some origin point. Like all vectors, a position vector can be represented by three coordinates that give the position of the object in the x, y, and z directions, respectively.

position vector

Magnitude of a Vector

Vectors, including position vectors, can also be represented by a magnitude, which tells you the total amount of the quantity the vector represents, and a second vector, called a unit vector, which only shows the direction and always has a magnitude of exactly one.

To find the magnitude of a three-dimensional vector, square each one of the coordinates, add them together, and find the square root of the sum, which you can see in the formula below.

vector magnitude

The magnitude of the position vector that we just saw would then be the following:

magnitude of a vector example

Unit Vectors

The unit vector, which is used to show direction, is found by dividing the three vector components by the vector's magnitude. The unit vector will always be in the same direction as the original vector and will always have a magnitude of one. As you can see here:

unit vectors

Multiplying the magnitude by the unit vector will give you the same three vector coordinates you started with.

unit vector back to coordinates

Equal vs. Parallel Vectors

How can you tell if two vectors are equal? For two vectors to be equal, they must have the same magnitude and direction. In component form, that means all three components of the vector must be the same.

equal vectors

Even if two vectors aren't equal, they might be parallel. If two vectors are parallel, they will never intersect each other. In order for this to happen, parallel vectors will have either the same direction or exactly the opposite direction. The magnitudes of two vectors that are parallel do not have to be the same.

parallel vectors

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