Calculating Magnitude & Components of a Vector

by Patrick Ford
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Hey, guys, let's work out this example together, we've got a vector. We're told what the white component is. We're also so what the angle is, and in this first part gonna find out what the magnitude of this a vector is. Let's check it out now. I always like to draw out the vector first. Feel like it's a great way to visualize what's going on. Make everything makes sense. So just draw out your little coordinate system here. We've got positive Why and positive X. Now we're told this vector has a white component of 12 but it makes an angle of 67. degrees. So start from the origin 67.4. Looks something like this doesn't have to be perfect. And we're trying to figure what the magnitude of this vector is. But if we break it down into its legs of the triangle, we know that the angle that it makes with the positive X axis is 67.4. And I know the why Components is gonna be 12 now in part B m actually going to figure out what the X component is. So that's unknown. So let's just get to our equations If I'm trying to figure out the magnitude of a how am I gonna do that? Well, I've got my vector composition and my decomposition equations over here. Remember, there's four equations that we know and the general rule is when we're going to come from components two vectors, we're gonna use our Pythagorean theorem in our tangent. Inverse, Where we're going from vectors down two components, we're gonna use our coastline data and a sign data equations. So let's just try to figure out the magnitude by using the Pythagorean theorem. What you're gonna find is that when you try to plug this in the magnitude, you're gonna end up with two unknowns because we know what our a white component is. But I don't know what my A X component is. In fact, that's actually what I want to be solving foreign part beat. And I'm also looking for the magnitude. So I've got two unknowns in this situation, so that means this equation here is not going to help us, so it might look like we're stuck, But there's actually other equations that we can use our vector decomposition or component equations that will also relate the components with the magnitude and the angle. So we're going to figure out which one of these equations we can use. Remember, there's four equations. So as long as we have two variables like a component and an angle or something like that, we can always use these equations to figure out the other two. So let's check it out. So I've got these equations here. My ex component is unknown. My magnitude is unknown, but my angle is known. So I've got two unknowns in here. I can't use that here, though. I've got my white components, I'm missing my magnitude. But I have the angle theta. So that means that this equation I can use to solve for the missing variable So my a y is gonna be the magnitude times the sign of the Data X. So that means that I could just move this over here. My my components 12. I'm looking forward. The magnitude right here s o. That means that then I've got to multiply by the sign of 67.4. And now if I just divide this over to the other side, I get 12 divided by sine of 67.4 is equal to a And if you work this out, you're gonna get 13 m as the magnitude of this angle or vector over here, it's gonna be 13 m. So now let's move on to part B, Part B. Now we have to figure out the X component of this triangle. So now we actually have three out of the four variables for our vector. So we can use actually any one of these equations to figure out this X component. So if our X component is missing, we can use any one of these equations. I always like to stick with the easiest one now that we like to Now that we actually have the magnitude of this vector, the easiest equation to use is gonna be our a co sign of data. But if you want, if you want to figure this out using the other equations, you are gonna get the same answer. So let's just use this equation. Are a X component is a times the co sign of theta X. So are a X Components is gonna be 13 times the co sign off 67.4 degrees. And when you plug this and you're gonna get five so one way you can also double check. This is you have now both components whips. So now we have both components, the five and 12. And if use the Pythagorean theorem, that should add up to 13. So that's one way you can kind of double check it anyway, guys, that's it for this one. Let me know if you have any questions.