### Video Transcript

We already know that a vector is a set of numbers that can be represented in a suitable space by a line segment with a specific length and direction. We can use them to represent forces or velocity or anything else thatโs got an amount and a specific direction associated with it.

In this video, weโre gonna look at the concept of unit vectors, which are basically line segments with the length of one unit pointing in a specific direction. Here weโve got vector ๐ด๐ต, which is four, three. This means that the journey from ๐ด to ๐ต involves of movement of positive four in the ๐ฅ-direction and positive three in the ๐ฆ-direction. So thatโs where these numbers come from here.

So what weโve ended up with here is a right-angled triangle because the ๐ฅ-axis and the ๐ฆ-axis are at right angles to each other, so this angle here is ninety degrees. So weโve got a little triangle and we really want to work out the length of this vector here, ๐ด๐ต, next, so weโre gonna use the Pythagorean theorem to work out that length, the hypotenuse of that triangle.

So bit of terminology, weโre gonna work out the magnitude of ๐ด๐ต; that is, the length of the vector ๐ด๐ต. And the notation that weโre gonna use for the magnitude of vector ๐ด๐ต is these vertical lines here, so thatโs the standard notation. So writing down the Pythagorean theorem, the magnitude of ๐ด๐ต all squared, so the length of the hypotenuse here all squared, is equal to the sum of the squares of the other side.

So ๐ด๐ต, the magnitude of ๐ด๐ต squared is equal to four squared plus three squared. And four squared is 16; three squared is nine; add those two together, we get 25. So the magnitude of ๐ด๐ต squared is 25. So if I now take square roots of both sides, that leaves me with the magnitude of ๐ด๐ต being the square root of 25, which is five. So weโve just worked out that the length of vector ๐ด๐ต is five units.

So if I wanted to generate a unit vector in the same direction as vector ๐ด๐ต, I can take vector ๐ด๐ต and just divide it by the length of itself. And the length of itself is the magnitude of the vector, so weโve got vector ๐ด๐ต divided by the magnitude of vector ๐ด๐ต, and we just worked that out to be five. So the unit vector in the direction of ๐ด๐ต is the vector ๐ด๐ต divided by five.

Now what that means is weโve got to take the ๐ฅ-component and divide it by five, and weโve gotta take the ๐ฆ-component and divide it by five. The ๐ฅ-component was four; the ๐ฆ-component was three. So the unit vector in the direction of ๐ด๐ต is this thing here: four-fifths is the ๐ฅ-component; three-fifths is the ๐ฆ-component. That will have a length of one.

So this red vector here, going from here to here, is the unit vector in the direction ๐ด๐ต. Itโs a fifth of the length of vector ๐ด๐ต. Itโs a vector which has an ๐ฅ-component of four-fifths and a ๐ฆ-component of three-fifths. Obviously I havenโt drawn that very accurately on here, but thatโs basically what the vector would look like. So the unit vector in the direction ๐ด๐ต, if ๐ด๐ต was four, three, is basically a vector which starts at ๐ด and has a length of one, but goes in the same direction that ๐ด๐ต would have gone in.

So in this case, weโve ended up with vector four-fifths, three-fifths. So letโs just check that in fact the length of this side here is one. So we can do the Pythagorean theorem again. So that length squared is four-fifths squared plus three-fifths squared, which is sixteen twenty-fifths plus nine twenty-fifths, which is twenty-five twenty-fifths which is one. So if I take the square root of that, that will give me the actual length, and the square root of one is equal to one. So that length is definitely equal to one. This is where the unit vector idea comes in.

So hereโs what the method is in general, and then weโll look at some more examples afterwards to make sure that you understand it. So letโs say weโve got a vector called ๐, which has an ๐ฅ-component of a and a ๐ฆ-component of ๐. Then the unit vector in the direction of vector ๐ is the vector ๐ divided by the magnitude of vector ๐. So whatever length ๐ has got, if I divide that length by itself, Iโm gonna get an answer of one. So this bit is forcing the length of that unit vector to have a magnitude of one.

And obviously itโs gonna still be in the same direction; the ๐ฅ- and the ๐ฆ-components will be in the same proportion as the original vector, so itโs gonna be heading in the same direction. So to work out the ๐ฅ-component, Iโve got the original ๐ฅ-component, ๐ด, and Iโm dividing that by the magnitude of the square root of ๐ด squared plus ๐ต squared, from the Pythagorean theorem. And to work out the ๐ฆ-component correspondingly, I take the original ๐ฆ-component and I divide that by the magnitude of the vector, the square root of ๐ด squared plus ๐ต squared. Thatโs gonna give me my unit vector in the direction of vector ๐. Now that might look a little bit intimidating when you write it all out like that, but itโs actually really easy to apply. So letโs have a look at an example.

So weโve gotta find the unit vector in the direction ๐ด๐ต, where ๐ด๐ต is the vector five, negative six. So the first thing is Iโm gonna do a little sketch to get this clear in my head. So it doesnโt have to be a hundred-percent-accurate diagram, but a quick sketch here gives you a sort of a flavour of whatโs going on. So weโve got the vector ๐ด๐ต; weโre going positive five in the ๐ฅ-direction, negative six in the ๐ฆ-direction.

The first thing we need to do is to work out the length of this vector ๐ด๐ต here. And then if we divide that vector by that length, itโs gonna create a little vector here, which starts at ๐ด, which goes in this direction, same direction as ๐ด๐ต, but itโll only have a length of one.

So Iโve got a right-angled triangle. So using the Pythagorean theorem, the magnitude of ๐ด๐ต, the length of a vector ๐ด๐ต, all squared is equal to the ๐ฅ-component squared, thatโs five squared, plus the ๐ฆ-component squared, thatโs negative six squared. So that is five squared is 25, negative six squared is 36. So we add them together, we get 61.

Now remember that was the magnitude squared. So taking square roots of both sides, Iโve got the magnitude of vector ๐ด๐ต is equal to the square root of 61. So we havenโt got quite such nice numbers this time, so weโll just leave that in that accurate format, root 61. And I can use that to work out what my unit vector in the direction of ๐ด๐ต is. Remember that would be the original vector ๐ด๐ต divided by the magnitude of that vector.

So our original vector was five, negative six. And weโve now got to divide each component by the magnitude of the vector. So five divided by the square root of 61 and negative six divided by the square root of 61. And thatโs it. This vector here โ five over root 61, negative six over root 61 โ has got a length of one. And the ๐ฅ- and the ๐ฆ-components are in the same proportion as the original vector ๐ด๐ต. In other words, itโs pointing in exactly the same direction, so itโs the unit vector in the direction ๐ด๐ต.

Okay letโs just do one more question then. Find the unit vector in the direction of ๐๐. So the difference here is weโve been given the point ๐, weโve been given the point ๐, and weโve got to calculate what the vectorโs gonna be.

So the first thing then is to sketch in our vector ๐๐, so it goes in that direction there, and weโve got to work out what the original vector ๐๐ is. So in the ๐ฅ-direction, weโre starting off with an ๐ฅ-coordinate of three, and weโre going down to an ๐ฅ-coordinate of negative five. So weโre going three here plus another five here. Thatโs negative eight in the ๐ฅ-direction. In the ๐ฆ-direction, weโre starting off at negative four and weโre moving. So weโre going four up to here and then another one up to here, so thatโs positive five in the ๐ฆ-direction.

So just summarising that, ๐๐ is negative eight, five. And now we can tackle this just like we did the previous question: use that triangle, do a bit of Pythagorean theorem to work out what the length of the vector ๐๐ is, and divide vector ๐๐ by that length. So the magnitude of ๐๐ squared is negative eight squared plus five squared, which is 64 plus 25. So remember thatโs ๐๐ squared; the magnitude of ๐๐ squared is equal to 89.

So taking the square root of both sides, and weโve ended up with another horrible number, so the magnitude of ๐๐ is the square root of 89. So Iโll just go through, divide the components of ๐๐ by root 89. And the unit vector in the direction ๐๐ is simply the original vector ๐๐ divided by the length of itself, so thatโs negative eight is the ๐ฅ-component divided by root 89, and five was the ๐ฆ-component, and divide that by root 89 as well.

And there we have it, the unit vector in the direction ๐๐. So weโve got some nice magic tricks we can use now, but why do we use unit vectors at all in the first place? Well, they encapsulate the information about the direction, but their length is just one. Theyโre like ready-made direction descriptors that you can multiply by a number to tell people how far to go in that direction. And it also turns out that theyโre useful in finding the angle between two different vectors, but thatโs a story for another day.

Okay we canโt resist it. Letโs just do one more example on unit vectors. Now you may recall, or hopefully you do recall, that weโve got special vectors ๐ and ๐, which are unit vectors in the direction of the positive ๐ฅ-axis and the positive ๐ฆ-axis. So ๐ is going to the right, and ๐ is moving up on the axes. So ๐ is one, zero and ๐ is zero, one. So if weโve got this vector here two ๐ minus three ๐, itโs just two of these ๐s strung together followed by the negative of three of the ๐s strung together.

In other words, thatโs two in the ๐ฅ-direction and negative three in the ๐ฆ-direction, but weโre not gonna worry too much about that format for this question because weโre gonna do it in terms of the ๐ and ๐ unit vectors. Now although weโve got ๐s and ๐s all over the place, the process is exactly what weโve just done before. So we want to work out the magnitude of the vector. And to do that, weโre gonna use the Pythagorean theorem.

Now two, positive two, is the ๐ฅ-component and negative three is the ๐ฆ-component. So we can just write that as the magnitude of this vector two ๐ minus three ๐ all squared is equal to the sum of the squares of the other side of the triangle. So thatโs two squared plus negative three squared, which is obviously four plus nine, which is 13. So remember that was the magnitude squared, so taking square roots of both sides says the magnitude of this line is the square root of 13.

And again, all we have to do to work out the unit vector in that direction is divide each of those components, the ๐ฅ- and ๐ฆ-components, by the magnitude of the resultant vector, so that was root 13. So two divided by root 13๐ and negative three divided by root 13๐. So even in the ๐, ๐-format of our vectors, the process is still just the same. To work out the unit vector in the direction of a particular vector, we take that vector and we divide by the magnitude of the vector. We basically divide each of the components in turn by the magnitude of the original vector, which we use Pythagorean theorem to calculate. So hopefully that gives you a basic insight into unit vectors.