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Resultants of Vectors: Definition & Calculation

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  • 0:02 What Is a Resultant of…
  • 1:25 Components & Resultants
  • 3:13 Example Calculation
  • 5:20 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 video, you will be able to explain what a resultant of a vector is and use mathematics to calculate the resultant of two vectors. A short quiz will follow.

What Is a Resultant of a Vector?

What is your velocity, right now? How would you answer that question? You might have answered super quickly, or you might have thought the question so difficult it's not even worth trying. Your response probably depends on how well you understand your place in the universe.

Right now, you're likely sat at a computer or tablet, so you might say your velocity is zero. But is it really? The earth is spinning at 465 meters per second and orbiting around the sun at 30,000 meters per second, and that says nothing for the motion of our solar system around the galaxy and the motion of our galaxy in the wider universe. You're always moving and in lots of different ways.

If we really wanted to figure out your total speed, you would have to take all of these velocity vectors and add them up to calculate a total velocity: an overall speed and direction. And this final result has a name. It's called the resultant of a vector.

The resultant of a vector is the total value after adding two or more vectors together. You can calculate a resultant using graphical methods. In fact, in another lesson, we go through the entire process of adding vectors using scale diagrams.

But we're not all wonderful at drawing scale diagrams, and humans are always prone to error. So in this lesson, we're going to talk about how to find the resultant of vectors using math: a quicker and more reliable way to do it.

Components and Resultants

There are two ways you might be given a vector. You might be told, for example, that a force of 20 newtons acts north-east or at 45 degrees to the positive x-direction. Just a number and a direction. You might, however, be given a vector in component form: the force acts 14.4 newtons north and 14.4 newtons east. This could also be written as 14.4i + 14.4j. (If this kind of notation is completely new to you, we talk about this component form in more detail in other lessons.)

This component form makes finding the resultant so much easier. Let's say you have to find the resultant of two force vectors: 3i + 4j and 2i - 6j. You just add up the forces in the x and the forces in the y. 3 + 2 = 5, and 4 + -6 = -2. So the resultant of those two vectors is 5i - 2j. Or in other words, 5 newtons in the positive x-direction, and 2 newtons in the negative y-direction. And if we're asked to also give our answer in component form, then that's it -- we're already done.

If we're not allowed to give our answer in component form, we can find the total using a combination of the Pythagorean theorem and SOHCAHTOA. Our two components are the adjacent (x) and opposite (y) sides of the triangle. So if we want the total force, we find the square root of 5 squared plus 2 squared, which is 5.39 newtons. And then you could use the sine, cosine or tangent equation (whichever you prefer) to figure out the angle, which turns out to be 21.8 degrees.

Example Calculation

Maybe an example would help. Let's say you're on one of those huge, wide moving walkways to help you get on a fairground ride. If the walkway is moving 3 m/s west and 4 m/s north, and you are walking 0.5 m/s west and 1 m/s south, what is your resultant (your total velocity) in component form? And what is the magnitude and direction of the resultant vector?

Well, all we have to do to get the velocity in component form is add the x values and add the y values separately. Let's call north to south our y-axis and east to west our x-axis. In the x-direction, we have 3 m/s west and 0.5 m/s west... that totals 3.5 m/s west. And in the y-direction, we have 4 m/s north and 1 m/s south... or in other words +4 and -1. Those total +3, which is 3 m/s north. So in component form, the total velocity is 3 m/s north and 3.5 m/s west.

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