Find the magnitude and direction of the vector represented by the...

Question

Find the magnitude and direction of the vector represented by the following pairs of components: (a) Ax = −8.60 cm, Ay = 5.20 cm:

Accepted Answer

Welcome back everybody. We are given the following pairs of X and Y components for two different vectors. And we are asked to find the magnitude and direction of those vectors. Now I want to give to helpful formulas for this task. We are given that the tangent of the angle with the positive X axis is equal to our Y component divided by our X component of a given vector. And we are given that the magnitude of the given vector is the square root of its X component squared plus its y component squared. With this in mind, let's go ahead and find the magnitude and direction for vector M. So the magnitude, we're just going to use this formula right here, M is equal to the square root of its X component 4.3 sq worse. Why component negative 8.6 squared When you plug that into your calculator you get 9. kilometers. Now let's go ahead and now find the angle from the positive X axis. So let's use this formula right here. And in fact what I'm gonna do to isolate data by itself is I'm going to take the arc tangent of both. Let's go ahead and do that. So we are going to get that the angle equal to the arc tangent of the y component divided by the exponents. So let's go ahead and plug those values in negative 8. divided by 4.3. This gives us the arc tangent of negative two. Which when you plug into your calculator, it's us negative 63.4 degrees. Now, is this the angle that we are looking for? Well let's go ahead and graph our vector. So it has a positive X component and a negative Y component. So we know it's going to be somewhere down here. Right. And so that's telling us that the angle that we found is this angle right here. However, the angle that we are looking for our theta M is going to be counterclockwise from the positive X axis. So it's actually this angle that we are looking for. So our data M is equal to 360 plus This angle right here which is negative 360 -63.4 is equal to 2 96.6°. So that goes ahead and takes care of our vector. Em let's go ahead and find and now using the same process. So ends magnitude is going to be the square root of its ex opponent which is negative 4.5 squared plus -8.7 Squared. When you plug this into your calculator, you get 9.79 m. Let's go ahead and try to find that angle. We're looking for our formula states that the arc tangent of its Y component over its X component negative 8.7 divided by negative 4.5. We get that this is the arc tangent Of 1.93 which is equal to 62.7°. Now, once again, is this the right angle? Let's go ahead and graph it. So we have a negative X. Opponent and a negative Y component, meaning that we are going to be somewhere in this quadrant, meaning the angle that we found. I'm going to be this angle right here. But we know that we need to have the counterclockwise angle from the positive X axis. So the data that we are trying to find is going to be degrees plus 62.72 gives us 242.7 degrees. With that we have found the angles of each of our vectors and their magnitudes corresponding to answer choice. A thank you guys so much for watching. Hope this video helped and we will see you all in the next one.

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