Welcome back, everyone. In previous videos, we've talked about how you can add or subtract vectors using the tip-to-tail method. Recall if we had one vector, we could add the tip of that vector to the tail of a second vector, and then the resultant vector that connected the two points would give you the sum of those vectors. This should be reviewed from previous videos. Now it turns out there is also a way that you can add or subtract vectors if you are given numbers as opposed to just a graph. And that's what we're going to be talking about in this video. This is a very important skill to have in this course as well as likely future math or science courses, so let's just go ahead and get right into things. If you have 2 vectors and you're given them in this component form, the way that you can add or subtract these vectors is by adding or subtracting the individual x and y components. To understand this, let's say that I have vector v, and I want to add or subtract it to vector u. What I just need to do is add or subtract the individual components. I can add or subtract the x components, so we'll have v_x ± u_x. And then, likewise, I can add or subtract the individual y components. And that's all there really is to adding or subtracting vectors. Let's go ahead and try it with this example down here. Here we have vector v = ⟨ 2, 3 ⟩, and we wish to add it to vector u = ⟨ 3, −1 ⟩. To do this, I'm first going to add the x components, 2 + 3 is 5, and then the y components, 3 + −1 is 2. So this right here is the vector v + u = ⟨ 5, 2 ⟩. That's the solution. If you use the tip-to-tail method on these vectors, you're going to find that you actually will get this result. If you have vector v = ⟨ 2, 3 ⟩ on a graph and vector u = ⟨ 3, −1 ⟩, and connect them tip to tail, you get exactly v + u. The math checks out visually as well. Another topic is multiplying a vector by a scalar, which is also crucial. Let's try an example where we have to find the vector v − 3u. First, we write out what we have. Vector v = ⟨ 8, 5 ⟩, and vector u = ⟨ 2, 4 ⟩. To find vector 3u, multiply each component of u by 3: 3*2=6 and 3*4=12, resulting in 3u = ⟨ 6, 12 ⟩. Then, subtracting 3u from v, 8 − 6 = 2 and 5 − 12 = −7. Therefore, v − 3u = ⟨ 2, −7 ⟩ is the final solution. Thanks for watching, and I hope you found this video helpful.