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Adding Vectors

Patrick Ford
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Hey, guys, let's check out this example together. So we've got these two vectors and A and B A has a magnitude of 10 and it's at degrees B has a magnitude three at 20 degrees, and we're gonna calculate the magnitude and direction off the resultant vector. But this result in Vector is not a plus B. It's a minus to be, so it's still gonna be vector addition. We just have to use a rules of subtraction and scaler multiples. So a minus to B is the same thing as saying a plus the negative of to be. So the first things first, we're just gonna draw and connect our vectors tip to tail. So we've got this vector A It's 10 at 40 degrees. So you start from the origin. So you've got this vector like this. This is gonna be a it's 10 and we know that this is a 40 degree angle. Now Victor B. Has a magnitude of three. The direction of 20. It's a little bit flatter. So we go from the tip to tail like this, and so this is gonna be RB vector and let's make a little bit shorter let's say that look like this. So we've got B is three. We know this angle here is 20 degrees s o. You know, hopefully you guys can see that. So now we have the vectors, we draw them tip to tail. But the problem is we have to remember that are resulting vector is not going to be from a to the end of B because we actually have to do a minus to be. So even though we've drawn out the vectors, they're actually still a little bit more than we need to do. So this result in vector first, we actually have to subtract to be here. So in order to subtract B, we're just gonna flip it. Basically, it's gonna go completely opposite direction with same length. So if be points in this direction, the negative B is gonna point exactly in the opposite direction. This is gonna be negative, B, and this is gonna be another negative be because we actually have to subtract to be so that means that my result in vector is again not from here to here. It actually you have to do a and you have to go minus to B and go in this direction. So my result in Vector is actually gonna be this vector over here. This is my our vector. So the second step here is we draw the resultant and its components. So that's the second step Rx and R. Why? So now we have to figure out the legs of this triangle over here. But, you know, again, So we have this vector here, we can calculate the components, but the resulting vector is actually if I go in this path over here, so let's go to the third step. So now, in order to calculate the legs of this, we have to break all the other triangles out into the components and then we have to calculate them. So this is my ex my A y. And then just for the sake of calculating the components of B X, we're gonna draw them out on this table over here or are in this diagram. So remember, we have to use a table in order to calculate all the X and Y components. So let's go ahead and do that. So I'm gonna get the table like this. This is my ex, and why components and then just for color coding, I'm just use This is my A and this is my B and then I'm gonna end up with a result in vector of R And then I have to write out the equation for our So let's go ahead and do that. So now if you want the X component of A we just have to stick to our vector decomposition equations. This is where we go to vectors from vectors, and then we want to decompose them into their components. So we just use our coastline data and a sign it data equations. So I'm gonna have for my ex component, I'm gonna use 10 and 40 degrees. And remember, as long as this is positive relative to the positive X direction and these equations gonna work So that means that my a co signed data here is gonna be 10 times the coastline of 40. When I end up with this 7.7, if you do the same thing for sign 10 times the sign of 40 degrees, what you get is 6.4. Now, if we do for B B cosigned data, that's this component over here So I know this is 7.7. This is 6.4. Might be cosine. Theta term is gonna be be times the cosine of 20 degrees. So this is gonna be three times the cosine of 20. And when I end up with his 2.8, so you know, this is 2.8 now, finally, three times the sign of 20 which is equal to one. So this is just one. All right, so these are all of our components. The table is really great at organizing all of this stuff. Now that links That brings us to the next step. If we want to calculate the result in vector, we're gonna have to combine all of these x and y components according to the our equation. I remember it's not a plus, B. You're not just gonna add all these components straight down, because remember, that is equal to a minus to be. What that means is that the X component of our is gonna be the X component of a minus two times the B X components. So if our is a minus to be, then Rx is a X minus two bx right just follows the same format. So my ex is gonna be 7.7 and then minus two times the X component, which is 2.8. If you work this out, your ex component is going to be 2.1. We do the exact same thing over here. This is gonna be a y minus to B y. So therefore, it's gonna be 6.1 minus two times one, and we get a white components of 4.4. So now we actually know what these are X and r Y components are. I know this is 2.1 and this is 4.4. And if you look at the diagram, it actually kind of makes sense. It's about 2.1 and 4.4. It's about twice as high. So this should make some sense to you. OK, so now let's last step. We just have to calculate our and tha r. And now we're going from the components of a vector. We have 2.1 and 4.4 the components, and now we actually want to construct the magnitude in the direction we use our Pythagorean theorem and our tangent inverse. So that means the magnitude of our it's just gonna be the Pythagorean theorem. We've got 2.1 squared, plus 4.4 squared. If you work this out, you're gonna get 4.9. So that is our results, since that's the magnitude of the results is now for the direction that's theta are This is gonna be the angle theta are over here. It's kind of messy, but I've got the tangent Inverse the arc tangent of my why component, which is my 4.4 over the X component, which is 2.1. So if you work this out, you're gonna get an angle of 64.5 degrees. So this over here, tha r is roughly 64.5 degrees. Alright, guys, that's it for this one. Let me know if you have any questions.