Vector Resolution: Definition & Practice Problems

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  • 0:02 What is a Vector Resolution?
  • 1:14 Example Calculation
  • 2:42 Solving the Example
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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 vector resolution is and solve problems involving vector resolution. A short quiz will follow.

What Is a Vector Resolution?

Vector resolution can mean a couple of different things, but it boils down to a process where one vector is broken down into two or more smaller vectors. This includes the process where a vector is broken into two components, which was discussed in much more detail in another lesson. But to summarize: a force of 50 newtons acting at 30 degrees above the horizontal could be described as 43 newtons upwards and 25 newtons sideways. By doing this, we've broken one vector into two smaller ones, like how 3 + 2 = 5, except in two dimensions. This allows us to do physics in the x-direction and y-direction separately, which makes problem-solving much easier.

But during this lesson, we're going to talk about problems where it isn't simply a horizontal and vertical component. We're going to talk about cases where the two vectors could be in any direction. A particular vector has any combination of smaller vectors that it could be broken into: after all, 1 + 5 = 6, but so does 2 + 4 and 3 + 3. Today, we'll go through a case where one of the numbers is missing, where 2 + BLANK = 6, and how you solve that when instead of simple numbers, you're dealing with vectors.

Example Calculation

Let's say that two kids are fighting over an ice cream cone. One is pulling north-east with a force of 50 newtons west and 50 newtons north, and the other is pulling a different direction with an unknown force. You might not know how much the other kid is pulling, but you do know what happens to the ice cream.

Well, I suppose the ice cream would actually most likely be squashed and fall on the floor. So, maybe that's a bad example. But let's assume that the ice cream stays intact - it's super-strength ice cream, and based on the way it's moving, you figure out that the overall force being applied to it must be 30 newtons west and 40 newtons north. This is a vector resolution problem because you could say that the total force being applied, F-total, is equal to the force applied by kid A, FA, plus the force applied by kid B, FB. But these are vectors, so it might help to show this as a vector triangle. These two forces add together to equal the total force. You could say that F-total is resolved into FA and FB.

To solve this problem, we need to figure out the two forces that, together, make up the total. We'll have to break down the total force into two smaller vectors that make it up. We already have figures for the total and we have figures for kid A. So, we just need to use those to figure out the force being applied by kid B.

If we write our forces in component form, we get these two values:

  • The force kid A is applying is 50 newtons east and 50 newtons north, so that can be written as 50i + 50j.
  • The total force, we said, is 30 newtons west and 40 newtons north, so that can be written as -30i + 40j.
    • By using 50i and -30i, I'm basically saying that east is positive and west is negative, which is what we'd get if we put a set of standard axis over a regular map.

Okay, so what are the missing values? Well, in component form this is super easy. Just look at one dimension at a time. 50i plus the i-value for the missing vector equals -30i. So, that must be -80i. And 50j plus the j value for the missing vector equals 40j, so that must be -10j. So, our missing vector is -80i - 10j. Or in other words, our missing force is equivalent to 80 newtons west and 10 newtons south.

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