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Standard Basis Vectors: Definition & Examples

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  • 0:01 What Are Standard…
  • 1:52 How Are Standard Basis…
  • 3:12 Examples
  • 5:08 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 standard basis vectors (otherwise known as unit vectors) are and how they are useful. A short quiz will follow.

What Are Standard Basis Vectors?

Think back to math class in school. That might be a long time ago for some of you, or just yesterday. But whenever it was, at some point in your many math classes, you probably used graph paper. And when you were very young, your graph paper was probably nothing more than a series of half centimeter boxes. Why were the boxes half a centimeter? Why not 1 centimeter? Or 1 millimeter? Or just some random size? For the purposes of mathematics, the size of the boxes is kind of arbitrary. It really doesn't matter.

But how would math class have gone if you hadn't had any boxes? Probably not very well. You need some size of box to compare one length of line to another, to keep things consistent. The same is true for unit vectors, otherwise known as standard basis vectors.

A standard basis vector is a 1 unit long vector that points directly in line with an axis. You could have an x-axis unit vector, which would be one grid square long in the x-direction and would look like this:

x axis unit vector on graph

Or a y-axis unit vector, which would be one grid square long in the y-direction and look like this:

y axis unit vector on graph

Or you could go into the z-axis, too, if you like making things more complicated. But these vectors basically represent the size of the boxes on your graph paper.

In physics, the size of the boxes we use tends to be 1 meter. We really like meters -- they're our standard unit for length. But whatever it is, we represent this size with an i for the x-direction unit vector, a j for the y-direction unit vector, and if we went into 3D, we'd use a k for the z-direction unit vector. They have a length of 1 unit each, whether 1 meter, 1 centimeter or 1 newton (for force).

How Are Standard Basis Vectors Useful?

Okay, but if the size of the grid squares is arbitrary, why do we have unit vectors? How are they actually used?

To be honest with you, if we get out of abstract mathematical theory and into the real world, the reason we use unit vectors is just that it looks neater.

Let's say you're re-arranging the furniture in your house, and you're pushing the couch down at an angle as you slide it across the floor, making a huge amount of noise and annoying the neighbors. You're pushing diagonally in two dimensions -- the x and the y. You could break that force into two components, and let's say, based on the angle you're pushing, you're pushing with 40 newtons in the x-direction and 10 newtons in the y-direction. 40 newtons forwards, 10 newtons down.

But writing '40 newtons forwards and 10 newtons down' is kind of inconvenient. We want a simpler way of writing it. So if we use 1 newton sized grid squares, we could say that your force vector is 40 i - 10 j newtons. That 10 is negative just because you're pushing down, and we tend to define up as positive. (Though that's actually arbitrary, too, as it happens.)

The important thing, though, is that this provides a nice way of writing out the value of a vector in component form. And that's the reason we use it in physics.

Examples

Let's put this idea into a couple of real life physics scenarios, to help us understand it.

One day you're driving your car. It's one of those fancy cars with all the bells and whistles. It also has a compass built in. That compass says you're heading directly northeast. And after converting into standard units, you figure out that you're traveling at 40 meters per second. Using some geometry to break that 40 figure into x and y components (as discussed in another lesson), you find that this is equivalent to 28.3 m/s east and 28.3 m/s north. It's perfectly balanced because you're going halfway between north and east. If we use the east-west direction as our x-axis and the north-south direction as our y-axis, that means that your velocity vector could be written as 28.3 i + 28.3 j meters per second.

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