 okay the wind is blowing outside but I want to talk to you about unit vectors let's have unit vectors very important concept as far as vector working with vectors is concerned now remember when we talked about vectors say a vector in two-dimensional space I have this vector here let's make it let's make it this is 4 and that is 3 on my y-axis and my x-axis so this is this point 4 comma 3 this is now my vector which I can write as 4 and 3 that is my vector V is my vector V so in vector notation column vector it's a point in a plane and we could work out the norm of this vector the length of this vector V normally like that and that is what the x value squared plus the y value squared if we had this in three-dimensional space if now these were all right angles to each right angles to each other and this was x y and z plane and we had this vector come out we would have x squared plus y squared plus z squared and we can take that up into a higher into higher dimensions as well so that was the norm of this vector but what if I have this vector this vector now this vector here is the addition of these two so if I had these two vectors and that's the beauty of these vectors I can deconstruct any vector according to these axes and later we'll see we don't actually have to use these axes very important concepts these I don't have to do that now let's get to this concept of the unit vector a unit vector is a vector with a norm of one a norm of one exactly one so the length of it is one so if this was four let's make that two one three so three two one it will have a norm of one and if this was one two three there that will have a norm of one now think about it this vector here along the x-axis this four is just so let's just look at this vector so let's look at wheels we'll just call this vector x for just to write something it will have along the x-axis a one and on the y-axis a zero and if I look at the x component of vector v I hope you can see down here if I look at the x component of vector v that is just four times one zero because that will give me this component here which is four zero four zero so it's a scalar multiplication and we haven't discussed that but it's this multiplication of each of these of this one with each of these that still gives me four zero so it's four along the x-axis nothing along the y-axis so there's something special about this unit vector because I can just multiply it by this scalar to get this one the same is going to happen there so the y component of v that is this zero and three but that is nothing other than three times zero and one and this zero and one is this unit vector in the y direction let's make it y hat and let's make this x1 x hat just to indicate that it's a unit vector some some books many books use this hat notation so that's this zero and one and if you were to just use the Pythagorean theorem you'd very quickly see that well the length of that must be one so what about this this vector yeah before we get that there's something something I want to subtly I want you to see if I have this plane and I have these two unit vectors I can add these two unit vectors let's make it x hat and y hat if I add those two so that we want zero plus zero one and that's going to give me one one one one one and not and that would be a point one one which will be up one one will be up there if I were to put a constant in front of this one and another constant in front of this one and this constant remember can be zero they could be negative I can get to any point in the plane you give me a point in this plane I can use these two unit vectors multiplied by a scalar because this one can get longer and shorter in this way this one can go this way in this way and I can get to any point on this plane and if I go to three dimensions I can get to any point in three dimensions if I had if I had another one if I had three dimensions there plus see three then and zero zero one there's something very special about these unit vectors they allow us to really get to the to really get to any way in a space there's something very special about them that I can utilize them to get anywhere in a space and we'll really look at that again in future lectures there's something very special about these unit vectors so that's a unit vector there as far as this vector and that vector is concerned but what about just the unit vector of this we can construct here also something that's of length one norm of one and if I were to multiply that by a scalar I'll get this original one so how do we go about that so that vector there that vector V was four and three and column notation if I were to multiply both with one over one over the norm of V think about it one over the norm of V one over the length of that so I'm dividing that each of these components by that so I'm taking this multiplying by that taking this multiplying by that so what is the norm of this vector well 3 plus 3 3 times 3 is 9 where are we so 4 squared is 16 and 3 square is 9 and that gives me 25 so the norm of this thing is 5 so that is going to be 1 over 5 and 4 and 3 so I'm going to get this four fifths and three fifths that is my unit vector and if you use the Pythagorean theorem you'll see that this equals one so it is this norm equals one just one and if I were to multiply this by five I would get this vector so I can deconstruct this vector into its unit vector of length one and to get to it I can just multiply by that so two very important concepts here one is to remember this equation for getting the unit vector of any vector it's one over the norm of that vector then times each of the components and the other thing is if I look at this how this vector is deconstructed along these two axes they are perpendicular they I can use this fact to get to any any point in the plane if it's two dimensions in 3d space if it's three dimensions and if it's higher I dimensions I can get to any point and we'll look at how special these unit vectors are and it doesn't necessarily have to be these we'll look at how we can construct these so I can still get to every point in the plane or in a space so in short that's a unit vector and the concept of getting to every point just by the use of a unit vector good let's have a look at unit vectors remember the norm of a vector if I were to have a vector and I would just to say the norm and let's keep it to what we had on the board a 4 comma 3 and we were to do that we'd see that the length of that is 5 and I if I were to take 1 over 5 and remember I can do that by control or command 4 slash just to get the fraction otherwise you could just say 1 divided by 5 let's do that just so just just for the sake of it I can put it in parentheses 1 divided by 5 that's going to be exactly the same thing and I going to multiply that by my vector which was 4 comma 3 and if I were to do that I get this vector 4 comma 4 over 5 and 3 over 5 and if I were to look at the norm of that vector and remember I can just use the percent symbol because that is what we had just executed and it's going to give me a norm of 1 so the norm of that vector is 1 that is a unit vector and if I were to say 5 times and let's put that that vector in 4 over 5 4 over 5 and 3 over 5 it's quite obvious what's going to happen here is that we are just going to get back to the initial vector so any initial vector can be let's say deconstructed into its unit vector which will have a length of 1 and just a scalar multiple to get back to that to that original vector and I'm back at the original vector of 4 of 4 comma 3 so that's one way to go about it I can also look at the function called normalize if I were to use normalize and I'm going to pass in that very same vector 4 comma 3 let's have a look at that and that will do all of this for me automatically I'm going to get the unit vector of that vector without any problems there is a function and it is called unit vector and let's have a look at how that works I'm going to say unit vector unit vector and very simply there are two arguments that I'm going to pass here first of all is the number of dimensions of my space let's say that it is 3 and I want the second unit vector in that 3 space and there we go at 0 1 0 as we said let's make a little table we're going to make a little table and we're going to make it of unit vectors unit vectors and let's make it in three-dimensional space comma I'm going to count them off n and we're going to make n go from 1 to 3 so I'm going to get the 3 unit and unit vectors inside of three-dimensional space they're going to be along the Cartesian plane x y in the z-axis and once we've done all of that let's see that I close all of my square that I all my square brackets are closed so let's just make sure of that unit vectors closed everything is closed and let's just have matrix form of this and if we look at the matrix form of this just concentrate on the three columns for now just the three columns I have 1 0 0 0 1 0 and 0 0 1 the norm so don't see it as a matrix at the moment it's just because we stated there that we wanted a matrix form just so that we have these column vectors but it is these three vectors that we can see the first one the second one and the third one hence there was our second one so unit vector will just give you this very easy way just to look at these they are quite simple if we just look at them orthogonal to each other just to do in your head obviously you're just going to bring the one down depending on you know which one of the unit vectors you're looking at so that's quite easy to do so there you go the unit vector you can do it by the norm and deconstructing it or just use the normalize function