Hey, guys. Let's check out this example together. We've got these 3 vectors, a, b, and c, and they're all written in their unit vector forms. And so we're going to calculate the magnitudes and directions of a bunch of these combinations of unit vectors. So let's check it out.

In the first part here, we've got to calculate magnitude and direction of d, which is just the addition of all the 3 unit vectors. So when we're doing unit vector addition, all we have to do is just line up the 3 vectors on top of each other. So let's write them out. My a vector is going to be 3 in the I, so it's basically 3 in the x minus 3 in the j, so it's basically 3 in the negative y direction. That's how you can think about that. My b vector is just going to be I minus 4j, and then my c vector we know is just negative 2i+5j. So if we want to calculate the resultant vector, this vector d here, by just adding all of them together, then I can just add up each of the components, basically all the parallel components, all the i's together, all the j's together, it's basically as if I already had the legs of all the triangles. I just have them in numbers. So I've got this d vector here, which is just a plus b plus c. So what I do is I just add these things straight down. So basically just add 3 and then the 1 and then the negative 2. That means that the x component of d is just going to be if I add all the components together. So 3 + 1 minus 2, that's going to be in the I direction. Plus, and now it's gonna be negative 3 minus 4 plus 5 and that's gonna be in the J. And so if I just add up all these numbers together, 3 plus 1 is 4 minus 2 is 2i and then I've got negative 3 and negative 4 is negative 7, plus 5 is negative 2. So that means I'm going to have 2i minus 2j and that is my D vector here.

But I'm not done because this is just the d vector written in unit vector form. So what I have to do is I actually have to get the magnitude and the direction of d. So let's just go ahead and sketch it out really quickly. What this would look like if you sort of like were to sketch it on a diagram, it doesn't have to be super pretty because we're just using it kind of as a sketch. We've got 2 in the x direction and then this minus sign is in the j direction. So we've got minus 2 in j. So that just means that we're going to go 2 in the x and then 2 in the y. So this means our vector is going to look something like this. This is 2 and then 2 like this. So we're going to calculate the magnitude and the direction of this vector over here. So d, and then I'll call this just theta d. So, my d is just going to be if I use the Pythagorean theorem, 2 squared plus negative 2 squared, since we've already got the legs of the triangle. And if you work this out, you're going to get 2.8. Now, the direction theta d is going to be if I do the tangent inverse and then I do the y component over the x component. But these strings are actually going to be the same. And by the way, this is technically supposed to be negative. So we've got, 2 over 2 because we're always plugging in the positive components. What you should get is you should get 45 degrees. And that makes sense because we've just got, like a perfect 45 degree angle, which means the components have to be the same. Whatever is over here is going to be over here. So that is our answer. We've got our magnitude and direction. So let's move on to part b now.

So we're going to calculate the magnitude and direction of a different combination, but it's the same principle. Here, we just have to use negative signs for when we add these vectors together. So let's see how that works. So we're going to do the same exact procedure here. So for b, we've got our a vectors. Actually, we might be able to just copy this over. So just go ahead and copy this over on your papers. Just like that. Okay. So now we're going to again, we're just going to be adding these things straight down. We're just going to be using a different formula. So instead of a plus b plus c over here, now this new vector which we're going to call e is actually going to be if we flip a and we flip b and then add it to c. So what happens when we actually make these vectors negative? Well, remember what happened when we did vector addition? We would just reverse the actual direction of those vectors? Well, all that's going to happen here for the legs of each of these triangles is that these vectors for a and b, we're going to have to reverse the sign of the components. So reverse the signs of components. So that just means that if we have instead of 3 I for our a vector in the x direction, we're going to have to use negative 3 because we have to basically just insert this little negative sign into that. So now, we got we got to flip the sign of the b vector which normally would just be 1, but now it's going to be negative 1. And then, we're going to add c. So we're just going to do nothing to c. It just stays the same. So now this is my new i direction. Plus, and now I've got to flip this over here. So if I flip this, this actually becomes positive. Then I've got to flip this one as well, so this also becomes positive. Right? Because I'm doing negative a minus b. And then I'm going to add it to 5 without doing anything. So that means that now my e vector, if I write it out, negative 3 minus 1 minus 2 is going to be negative 6 in the I, and 3 plus 4 plus 5 is actually 12. So now my I've got negative 6 I plus 12 j, And this is my e vector here. So I can do the same thing. I can basically just sketch out what this would look like. And if I have this little diagram like this, then this vector would look like Well, I have a component that's negative, so it points to the left. So that means that my x component here would look like this. And I know that this ex is equal to negative 6, and then I have my which is equal to 12. It doesn't have to be necessarily to scale, but this is what our vector would look like just to get an idea of what's actually going on. So we want to calculate now the magnitude and directions. We're going to use the same equations that we did before. So if we want to calculate the magnitude of e, we just use the Pythagorean theorem. So you just use the Pythagorean theorem of 6 squared or actually negative 6 squared, but it doesn't matter because you always get a positive number. And then 12 squared, and you get 13.4. So that is the magnitude. As for the direction, we're going to get the tangent inverse. And now, oops. So tangent inverse, and this is going to be the y component which is 12 over 6, and always going to be positive numbers, and you're going to get 63.4 degrees.

Alright, guys. That's