Hey, guys. So previously, we saw how to add vectors graphically. A situation like this where you have vectors a and b and you want to calculate the result, so you add them tip to tail and then you count up a bunch of boxes. Well, sometimes, unfortunately, you're gonna have to calculate the resultants without being able to count squares because you're gonna have diagrams that don't have grids. So in this video, we're gonna see that all we just need are all of our vector math equations or triangle math equations, and we're gonna pull them all together to solve these kinds of problems. Now these problems, as we're gonna work one out together, are actually really, really repetitive. You end up doing the same things over and over and over again. So I'm gonna give you a list of steps. It's gonna help you get the right answer every single time. Let's check it out.

So this problem says that we're walking 5 meters at 53 degrees, so that's a displacement, and then another displacement in another direction. We're gonna calculate the magnitude and the direction of the total displacement. Remember, total displacement can just be thought of as a resultant vector. So we're gonna be calculating a resultant vector. Alright. So the first step we're gonna do in these problems is we're gonna draw and then connect the vectors tip to tail. So when we were given them graphically, we didn't have to draw them, but now that we don't have them, we're gonna have to draw them out. And because we don't have a grid, we're just gonna kinda sketch them out to the best of our ability. So we're gonna start from the origin, 5 meters at 53, looks something like this. That looks about right. So this is 5. We know this direction here which is relative to the x-axis is 53 and we've got 8 at 30. So connect them tip to tail and this vector here, 30 degrees is a little bit shallower than 53, a little bit flatter, so it's gonna look something like this. So it's 8, doesn't have to be perfect and we know that this is 30 degrees above the x-axis. Cool.

So that's the first step. The second one is just we have just drawn out what the resultant vector is gonna look like. So what is the resultant? Well, when we did this graphically, we just connect them tip to tail and the resultant was the shortest path from start to finish. It's basically as if we had actually just walked in this direction. The principle is the exact same. It's basically the shortest path from the start of the first one to the end of the last and this is gonna be my result in vector. So this is gonna be the magnitude r and then it's specified by an angle which is theta r.

So we're trying to figure out the magnitude and the direction, which means that we're gonna need the components. Remember how we how do we solve for this? We take this result in vector and break it up into the triangles and we have to figure out what Rx and Ry are. So we have to get these components over here. When we did this graphically, this was pretty straightforward. We have the magnitude, we break it up into its little triangles, we get r x and r y, we could just count up the little boxes here. It was pretty straightforward. This was just 4 and this is also just 4. We don't have boxes in this situation, so we can't calculate them by just adding up all the boxes. There's nothing for us to count.

We're gonna need a new method to figure out what these components are. So one way to think about how we got this Rx component is we can kinda break up the smaller vectors a and b into their small little triangles, so we can kinda think about this little triangle up here, this little x component. Basically, we just think about this a vector as breaking up into its triangles and we have this ax component and then we also have bx component as well. And so we know this ax component is 3, we just count the boxes, you know, this bx component is just 1 and so one way you can think about this is that this 4 is really just the addition of this 3 and this one put together.

Now, this is gonna work the exact same way over here. So what we have to do is we have to break up each of the vectors into the triangle. So this is my ax and this is my ay and then this is going to be my bx, and this is gonna be my by. So now, that leads us to the 3rd step, which is we have to calculate what all of these components are. So that brings us to our equations. How do we take a vector which we have the magnitude and the direction and calculate the components? We're just gonna use our vector decomposition equations. So that's which Our a cosine theta and our a sine theta equations. Now, we have 2 vectors. This is a and this is gonna be b, so the best way to keep track of all of the components is, oops. I'm sorry. I'm supposed to make that green.

So the best way to keep track of all these components is by building a table. You're gonna have a lot of stuff floating around in your papers everywhere. It's good to get organized, so I always highly encourage that you guys build the table. So we want the ax components and the ay components. So this we're just gonna use our a cosine theta and a sin theta equations. So my a vector, the magnitude is 5, so this is gonna be 5 times the cosine of 53 degrees and we're gonna get 3. If you do the same thing for the y direction you use sin...