Worked example: finding unit vector with given direction
What I want to do in this video is explore the idea of a unit vector. A unit vector is just a vector that goes in a particular direction that has a magnitude of one. Let's take an example. Let's say that I have the vector, let's say the vector A, and in the horizontal direction for every three that it moves in the vertical direction it moves up four. What else do we know about this? We could figure out A's magnitude, we can denote it like this. The magnitude of vector A, well this would just be the length of it. Let's try to visualize A. For every three that we go in the horizontal direction we're going to go four in the vertical direction. Vector A would look something like this. It would look like that. That is vector A. What's its magnitude? The magnitude is just the length of this vector right over here and we can use the Pythagorean theorem to figure this out. This length is going to be the square root of the sum of the squares of the other two sides. This is just the hypotenuse of this right triangle. This is going to be three squared plus four squared or this is going to be the square root of nine plus 16, square root of 25, or it's going to be equal to five. You might have just recognized that this would be a three, four, five right triangle. The length of this side right over here is five so I could say the magnitude of A is equal to five. This is clearly not a unit vector. It has a length, it has a magnitude of something other than one. Let's say we wanted to construct a unit vector that has the same direction as A but has a length of only one. Another way of thinking about it, let's say we wanted to figure out a vector that goes in the exact same direction but it has one fifth the magnitude, it only has a magnitude of one. What could we do? If we scale everything down by a fifth, if we were to multiply each of these components of our vector A by a fifth, or another way of thinking about it, if we divide each of these by the magnitude of A then we can construct this unit vector. I'm going to call that unit vector, I'll call it A but instead of putting an arrow on top I'm going to put -- Actually just to not confuse ourselves let's call it U for a unit vector. To also make it clear it's a unit vector and not just a normal vector I'm going to put this little hat. Instead of this little arrow when you put this hat this denotes that you're dealing with a unit vector, a vector with magnitude of one. The unit vector, we could write it down. It's going to be equal to each of these components, for A we just divide by the magnitude of A. It's three in the horizontal direction, four in the vertical direction. Once again we just divide by the magnitude, magnitude of our vector. This is going to be equal to, the magnitude we already figured out is five so it's going to be three fifths in the horizontal direction and four fifths in the vertical direction. You can verify. This is going to have, one way to think about it the ratio between these two numbers is the exact same thing as the ratio up here so we're going in the same direction but the magnitude here is now going to be equal to one. You can verify that, let's do that. What's the magnitude now of our unit vector? Let me write that hat a little bit, that hat got a little crooked. What's the magnitude of unit vector U? It's going to be the square root of the sum of the squares of these two components. It's going to be the, the square of three fifths is nine over 25 plus the square of four fifths is 16 over 25. What's that going to be? Well nine plus 16 is 25 so it's going to be the square root of 25. That plus that is 25. 25 over 25 which is just going to be equal to the square root, or I guess we're doing the principal root, the positive square root of one, which is just equal to one, which is exactly what we wanted. It goes in the same direction but magnitude is one, that's why it's a unit vector.