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Vector Components: The Magnitude of a Vector

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  • 0:01 What Is a Vector?
  • 0:57 The Magnitude of a Vector
  • 2:13 Example
  • 2:55 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 the magnitude of a vector refers to, and how to calculate it when given the components of a vector. A short quiz will follow.

What is a Vector?

A vector is a quantity that has both magnitude (numerical size) and direction. For example, 20 miles per hour (speed) is not a vector, whereas 20 miles per hour north (velocity) is a vector.

Vectors can point in any direction, and will often be directed diagonally in two dimensions, the x and the y direction. Because of this, it's often useful to break it into components, as described in detail in another lesson. In component form, you might be told that a beach ball is feeling a diagonal force that's equivalent to 150 newtons left and 50 newtons up, for example. You might also see this written with unit vectors in the form 150i + 50j. But however it's written, you might need to go the other way - you might need to take those components and figure out the overall magnitude of the vector.

The Magnitude of a Vector

The magnitude of a vector is just the numerical size. So if you shoot a cannonball at 30 degrees to the horizontal, at a speed of 50 meters per second, the magnitude is 50 meters per second.

If, instead of knowing that overall magnitude of 50 meters per second, you know the two components, in this case 43.3 meters per second sideways, and 25 meters per second up, there has to be a way to figure out that overall magnitude.

A vector triangle is used to determine magnitude
Example of a vector triangle

When breaking a force into components, you create a vector triangle like this one and use SOHCAHTOA geometry to figure out the other two sides of the triangle - the x-component and y-component. That's your 43.3 meters per second and 25 meters per second. Again, this is discussed in full detail in another lesson. But if you don't know the hypotenuse of the triangle, how can you calculate it?

Well, just like any right-angled triangle, if you know the adjacent and opposite sides, you can use the Pythagorean theorem to find the longest side (the hypotenuse). So in this case, the magnitude of the velocity vector squared is equal to 43.3 squared, plus 25 squared. Take the square-root of both sides, and you get 50 meters per second, which is what we'd expect. And that's really all there is to it.

Example

So let's say you're dragging a kitchen table along the floor. You're pulling up with a force of 25 newtons, and sideways with a force of 80 newtons, and you're asked for the overall magnitude of the force you're applying.

First, we write down what we know. The force in the x-direction, Fx, is 80 newtons, and the force in the y-direction, Fy, is 25 newtons. We can make a vector triangle out of that force - you're pulling with 80 newtons this way, and 25 newtons this way. And the hypotenuse of the triangle is what we need to calculate - the overall magnitude.

Plug those numbers into the Pythagorean theorem, take the square root of both sides, and solve. And you get 83.8 newtons. And that's it; we're done.

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