Welcome back, everyone. So in this video, we're going to be talking about one of the more critical topics both in math, as well as future science courses you will likely take, which is something known as vectors. Now to understand vectors, I want you to imagine that two people walk up to you. The first person tells you, hey, I just ran 6 miles per hour. And the second person tells you, I just ran 6 miles per hour northeast. What is really the difference between what these two people told you? Well, the first person gave you a magnitude. They told you how much they were running. Now the second person gave you a magnitude, but they also gave you a direction. They told you where they were running. And this second person gave you an example of a vector. A vector is a quantity that has both magnitude and direction.

Now to understand mathematically how these vectors are represented and what they look like, I think it's best if we just jump into some examples of a vector. So vectors visually are represented by arrows. So if I were to take an arrow and draw it between these two points, this arrow would be an example of a vector. Now typically, you're going to see vectors written like this, where we have this v with a little arrow on top here. It's also possible that you will see notation like that. It really just depends on the textbook you use and what class you're in, but all of these are examples of the same thing which is this arrow known as a vector. Now where the vector starts, this tail end of the vector here, this is what we call the initial point. And where this vector finishes, where the tip of the arrow is, this is what we call the terminal point. And I think this makes sense because, naturally, if you draw this vector, you're going to draw it from initial to terminal point when you put this arrow here.

Now the length of the vector is actually very important, because the length tells you the magnitude of the vector. So let's just use this same velocity that we have up here. If this vector has a magnitude of 6 miles per hour, then that tells you how long this vector is. Now let's say I was Superman, and I was running 50 miles per hour. In that case, I would need a significantly longer vector. Whereas if I was only walking 1 mile per hour, I would need a shorter vector. So that's the idea of a vector's magnitude.

Now the direction of a vector is dependent on the angle of the vector. Let's say that this vector that we have here has an angle of 30 degrees as it points northeast. This would be how you could find the vector's direction. So as you can see, vectors are actually pretty straightforward. All they really do is give us a direction to these quantities or magnitudes that we've learned about throughout math.

Now something else that's quite interesting about vectors is it's actually possible for vectors to be negative. Now this might sound confusing at first, because what would it really mean if I was running negative six miles per hour? That doesn't seem to make a ton of sense. Well, it actually turns out that all a negative vector really does is causes the initial vector that you have to point in the opposite direction. It has the same magnitude, but opposite direction. So let's say that we had negative v. If our vector v is 6 miles per hour, negative v would be negative 6 miles per hour. And what this would mean is that our vector which is initially pointing northeast is now going to point southwest. So in this instance, we would now have the initial point B at point B, and the terminal point b at point a.

Now one more thing I want to cover before finishing this video is another case which is known as the 0 vector. The 0 vector has a magnitude of 0 and no direction. And I think that this makes sense. Because if I told you I was running 0 miles per hour, well, 0 miles per hour means I'm not moving at all, I'm just holding still. And if I'm holding still, there's no direction I'm traveling in. So it would make sense that I would have no direction and no magnitude.

So this is really the main idea of vectors and how they're simply quantities that have both magnitude as well as direction. So I hope you found this video helpful. Thanks for watching.