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UExcel Physics: Study Guide & Test Prep17 chapters | 188 lessons

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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 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.

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.

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.

Now to figure out the total magnitude and direction. We have a right-angled triangle that represents 3.5 m/s west and 3 m/s north. Using the Pythagorean theorem, we can find the hypotenuse of the triangle, the total magnitude of the velocity: this will equal the square root of 3.5 squared plus 3 squared, which is 4.6 m/s. And we can find the direction - the angle - by taking the inverse tangent of the adjacent side divided by the opposite side, which is the inverse tangent of 3 divided by 3.5. That gives us an angle of 40.6 degrees above the negative *x*-axis.

So our resultant velocity - our total velocity - is 4.6 m/s at 40.6 degrees above the negative *x*-axis. (Or equivalently, 3 m/s north and 3.5 m/s west.) And that's it! We're done.

The **resultant of a vector** is the total value after adding two or more vectors together. You can calculate the resultant using graphical methods, but you can also do it using equations. You might want to do this if you have several velocities to combine together, like your velocity across the earth, the rotation of the earth, and the movement of the earth around the sun, for example.

To find the resultant of two vectors in component form, just add the *x* components of each and the *y* components of each. If you're given the total magnitude of a vector and the angle it's pointed at, you'll need to break it into components first (as discussed in another lesson).

Once you have your resultant vector in component form, you can convert that final vector into a total magnitude and direction using a combination of the Pythagorean theorem (to get the magnitude) and SOHCAHTOA (to get the direction).

Finding the resultants of vectors can vary from incredibly easy to absurdly hard, depending on the exact situation, the information you're given, and the form of the answer you need to provide. It requires a wide and full knowledge of vectors to solve some of these problems. But as long as you act carefully and methodically, even the most complex problems can be solved.

This lesson can be used to enhance your ability to:

- Determine what the resultant of a vector is
- Find the resultant of two vectors in component form
- Illustrate the method used to convert the resultant vector in component form to the total magnitude and direction

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UExcel Physics: Study Guide & Test Prep17 chapters | 188 lessons

- What Is a Vector? - Definition & Types 5:10
- Vector Addition (Geometric Approach): Explanation & Examples 4:32
- Resultants of Vectors: Definition & Calculation 6:35
- Vector Subtraction (Geometric): Formula & Examples 5:48
- Standard Basis Vectors: Definition & Examples 5:48
- How to Do Vector Operations Using Components 6:29
- Vector Components: The Magnitude of a Vector 3:55
- Vector Components: The Direction of a Vector 3:34
- Vector Resolution: Definition & Practice Problems 5:36
- Go to Vectors

- Go to Kinematics

- Go to Relativity

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