Magnitude of vectors
Vector magnitude from components
- [Voiceover] Let's do some examples figuring out the magnitude of a vector if we're just given some information about it. So, one of the simplest cases would be well, if they just told us the actual components of the vector. So if they said vector a is equal to, let's say five comma negative three, this means that its x-component is positive five, its y-component is negative three. Well, if we have this, then the magnitude of a, the magnitude of a is just going to be, and this really just comes from the distance formula which just comes from the Pythagorean theorem, the magnitude of a is just going to be the square root of the x-component squared. So let me do that in a different color. So the square root of the x-component squared, so five squared, plus the y-component squared, so plus negative three squared. And this is going to be equal to the square root of 25, 25 plus nine, plus nine, which is equal to the square root of 34, which is equal to the square root of 34. And if you want to think about this visually, this is very easy to do just looking at the actual components. But if you want to make sense of this, why this is essentially just the Pythagorean theorem, we could draw out a quick coordinate axis right over here. So that's our y-axis. This is our, let's see, I have a y-component of negative three. So let's see. That is our, actually let me draw it a little bit different. Let me draw it like this. That is our x-axis. And we see its x-component is positive five, so one, two, three, four, five. That's five there. And its y-component is negative three. So one, two, three. And so this is negative three. And so we can draw this vector with its initial point. Remember, we can always shift around a vector as long as we don't change its magnitude and to direction. We can start it at the origin, and make it go five in the x-direction and negative three in the y-direction, and so its terminal point will be right over there at the point five comma negative three. And so the vector, the vector, will look like this. And if we want to figure out the magnitude, that's just the length of this line. And what we can do is just set up a right triangle where our change, our change in y is this negative three right over here. That is our change in y. And our change in x is this positive five, is that positive five. And so this is a right triangle. Five squared plus, you could just view the absolute value of this side as three, so five squared plus three squared is going to be the hypotenuse squared. Comes straight out of the Pythagorean theorem.