For the vectors A and B in Fig. E1.24 use the method of components to find the magnitude and direction of (d) the vector difference B - A

Video transcript

welcome back everybody. We are given two vectors and we need to find d minus C. Well, I'm actually gonna think about this a different way. I'm going to think about this as D plus the quote unquote reverse of C. But what does this reverse look like? Well, I'm actually going to draw it here. It is going to have the same magnitude at sea but it is going to be in the other 180 degree direction. Right? So what is this going to look like? Well, it is going to look something like this. This is going to be our negative C. Now we're gonna add D plus negative seeds. So let's go ahead and do that on our graph. So we have our D. Right here. I'm gonna add negative C. By starting out where D ends and then adding the vector negative see to it. Now using the tip to tail method, if we draw a vector from where we started which was the origin to the end of negative C. Which I just let me label that real quick just to make sure we don't get lost here. This is going to be our D. Plus negative C. Which I'm going to call are now we are asked to find the magnitude and angle two of our. Now please keep in mind that this is not going to be the scale and that will be important in the future here. Okay, so if we're gonna find the magnitude and direction of a vector, we need to know some important formulas. So the magnitude of a given vector, let's say a is going to be the square root of its x component squared plus its y component squared in the direction that we are looking for is going to be represented by the angle Theta is going to be equal to the arc tangent of its Y component over its X component. But what are the X and Y components of our? Well, think about it this way. Right, your R X is just going to be equal to the X component of D plus the X component of the reverse of C. Same thing with your Y component, you are going to have that you're Y component is equal to the Y component of D plus the Y component of the reverse of C. So let's go ahead and find our X and our Y. So our X is equal to D X. Well, what is D X here? Well, we are looking Or this stretch right here. So DX is going to be the magnitude of D which is Times The sine of that angle 60 and it is going to be negative since we are going in this direction plus the X component of negative C. Well, here's an important thing to observe since C is a complete 1 80 degree turn, we have a set of two parallel lines with that being said since this angle is 50 degrees. We are also going to have that this angle is 50 degrees which helps us out. So now we are going to try to have to find this guy, this is going to be our negative C. X. So this is gonna be equal to the magnitude of C. Which still has the same magnitude Times The Co Sign of 50. When you plug this into, your calculator is equal to negative 4.41 m. Right? So let's go ahead and find our Y. With the same method. Dy we are looking at this guy right here, going to be the magnitude of D 14 this time times the co sign of our angle and it will be positive since we are going in the positive Y direction plus the Y component of our negative C. Y. So this time we're looking at this guy and this is going to be the magnitude of C, which is 12 times a sign of our angle 50 plug this into your calculator, you get 17.39 m now that we have our X. And our Y. We are ready to find the magnitude and direction of our So the magnitude of our is going to be equal to the square root of its X component which is negative 4.41 squared plus 17.39 Squared. And when you plug this into your calculator, you get 17.94 m. Now let's go ahead and find the direction. Now remember earlier I said it's not the scale. so it'll look a little wonky but you'll see what I mean just a second. So our angle theta is equal to our arc tangent of our wide component of our exponents. So we have negative 4.4 1/ 0.39. This when you plug it into your calculator is going to be negative 14. degrees. But is this the angle that we are looking for? What this is saying is that we are 14. degrees away from the negative X axis. So this is the angle that we found. But what we really need is we need this angle right here which is the counterclockwise angle from the positive X axis. What we're gonna do is we're going to do data is equal to 1 80 minus this angle right here minus 14.23. And this is going to give us a final answer of 165.77 degrees. So now we have found the magnitude of our d minus C vector and we have found the direction. Giving us a final answer of answer choice. C. Thank you guys so much for watching. Hope this video helped. We will see you all in the next one