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:
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.
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.
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.
The magnitude of the position vector that we just saw would then be the following:
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:
Multiplying the magnitude by the unit vector will give you the same three vector coordinates you started with.
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.
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.
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To determine if two vectors are parallel, you need to know their unit vectors. Since the unit vector is the part of the vector that shows you its direction, parallel vectors will have the same unit vector.
How can you tell if the two vectors shown below are parallel or not?
To know if they're parallel or not, you need to find the unit vector for each one. We already know the unit vector for the first one. To get the unit vector for the second one, first find the magnitude of the vector and then divide the original three components by the magnitude. As we can see played out in the formula below, just like last time, we find:
As you can see, we get the same unit vectors as before. Because Vector A and Vector B have the same unit vector, it means they're in the same direction and are, therefore, parallel.
Certain parallel vectors may also be collinear if they lie on the same line. Other vectors can be coplanar as long as they lie on the same two-dimensional plane, even if they aren't parallel.
A vector is a quantity with both a magnitude and a direction. One important type of vector is a position vector that gives the position of an object relative to some origin point. A position vector, and all other types of vectors, can be represented graphically with an arrow, or mathematically with three coordinates or a magnitude and unit vector. The magnitude tells you the total amount of the quantity the vector represents, and the unit vector only shows the direction and always has a magnitude of exactly one.
For two vectors to be equal, they must have the same magnitude and direction. 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. Certain parallel vectors may also be collinear if they lie on the same line. Other vectors can be coplanar as long as they lie on the same two-dimensional plane, even if they aren't parallel.
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