Learn about Unit Vector Notation

Khan Academy Presents: Expressing a vector as the scaled sum of unit vectors

Good afternoon! We have done a lot of work with vectors and a lot of the problems when we launched something in the projectile motion problems or when we were doing the inclined plane problems. We always broke, I always gave you a vector like I would draw a vector like this. I would say some things has a velocity of 10 m/s, it is at a 30-degree angle and then I would break it up into the x and y components. So, if I called this vector V, I would use a notation Vx and you know the Vx would have been this vector right here. Vx would have been this vector down here, the x component of the vector. And then Vy would have been the y component of the vector and it would have been this vector. So this was Vx, this is Vy right? And hopefully by now, it is like a nature of how we would figure these things out. Vx would be 10 times cosine of this angle, 10cos30 degrees which I think is square root of 3/2 but we are not worried about that right now and Vy would be 10 times the sine of that angle. This hopefully should be second nature to you. If it is not, you can just go through soh-cah-toa and say well, the sine of 30 degrees is the opposite of the hypotenuse and you would get back to this but we have reviewed all of that, you should review the initial vector videos. But what I want you to do now because this is useful for simple projectile motion problem but once we start dealing with more complicated vectors and maybe we are dealing with multidimensional vectors, three dimensional vectors or we start doing Wiener Algebra where we do n-dimensional vectors, we need a coherent way of, an analytical ways of having to always draw a picture of representing vectors. So, what we do is we use something I call and I think everyone calls unit vector notation so what does that mean? So, we define these unit vectors. Let me draw some axis. It’s important to keep in mind. This might seem a little confusing at first but this is no different than what we have been doing in our Physics problem so far, so let me draw the axis, draw the axis right there and let us say that this is one, this is zero, this is x, what am I doing, this is two 0,1,2 I do not know I must have been writing in Arabic or something going backwards but this is 0,1,2 that is not the twenty. And then let us say this is one, this is two and the y direction. I am going to define what I called the unit vectors in two dimensions. So, I am going to first define a vector. I will call this vector i and this is the vector. It just goes straight in the x direction, it has no y component and it has a magnitude of one. And so this is i and we denote the unit vector by putting this little cap on top of it. There is multiple notations, sometimes in a book you will see this i without the cap and it is just bold face. There is some other notations but if you see i and not in the imaginary number sense you should realize that that is the unit vector. It has magnitude one and it is completely in the x direction and I am going to define another vector and that one is called j and that is the same thing but in the y direction. That is the vector j and you put a little cap over it. So, why did I do this? Well, if I am dealing with two dimensions and as later we we’ll see in three dimensions so it will actually be a third dimension and we will call that k but do not worry about that right now but if we are dealing in two-dimensions, we can define any vector in terms of sum, sum, sum of these two vectors. So, how does that work? Well, this vector here let us call it v right? This vector v is the sum of its x component plus its y component, right? When you add vectors, you can put them head to tail like these and that is the sum. So, hopefully knowing what we already know, we knew that the vector v is equal to its x component, its x component plus its y component. When you add vectors, you essentially just put them head to tails and then the resulting sum is kind of where you end up, right? So, it would be, if you added this vector and then you put this tail to this head then you end up there so you end up there so that is the vector. So, can we define Vx as sum multiple of i of this unit vector? Well sure, Vx completely goes in the x direction right? But it does not have a magnitude of one; it has a magnitude of 10cos30º. So, its magnitude is 10 so this is—let me draw the unit vector up here. This is the unit vector i. It is going to look something like this and this. So, Vx is in the exact same direction and it is just a scaled version of this unit vector and what multiple is it of that unit vector? Well, the unit vector has a magnitude of one. This has a magnitude of 10cos30º so I think that is what like five square roots of three or something like that. So, we can write Vx, we can write Vx is equal to 10cos30º times the unit vector i. Let me stay in that color so you do not get confused times the unit vector i. Does that make sense? Well, the unit vector i goes in the exact same direction but the x component of this vector is just a lot longer, it is 10cos30º long and that is equal to cos30º is square root of 3/2 so that is 5 square root of 3i. Similarly, we can write the y component of this vector as sum multiple of j so we could say Vy, the y component, well what is sine of 30º? Sin30º is 1/2, so ½*10, so this is 5. So the y component goes completely in the y direction so it is just going to be a multiple of the unit vector j and what multiple is it? Well, it has length five while the unit vector has just length one, so it is just five times the unit vector j. So, how can we write the vector v? Well, we know the vector v is the sum of its x component and its y component. And we also know, so this is the whole vector v, what is its x component? Well, its x component can be written as a multiple of the x unit vector, that is that right there, so you can write it as five square roots of three i plus its y component, so what its component? Well, its y component is just a multiple of the y unit vector which is called j with a little funny hat on top and that is just this; it is 5*j. So what we have done now by defining these unit vectors and I could switch this color just to so you remember that this is i, that this unit vector is this. In using unit vectors in two dimensions and we can eventually do them in multiple dimensions, we can analytically express any two-dimensional vector instead of having to always draw it like we did before and having to break out its components and always do it visually, we can stay in kind of analytical mode and non-graphical mode and what makes this very useful is that if I can write a vector in this format, I can add them and subtract them without having to resort to visual means. And what do I mean by that? So, if I define some vector A is equal to 2i+3j and I have some other vector, this is a vector, and I have some other vector b. This little arrow just means it is a vector. Sometimes you will see this as a whole arrow—10i plus 2j. If I were to say what is the sum of these two vectors, a+b? Before we had this unit vector notation, we would have to draw them and put them heads to tails and it did not get to do it visually and it would take you a lot of time but once you have it broken up into the x and y components, you can just separately add the x and y components so vector a plus vector b, that is just (2+10)*i +(3+2)*j and that is equal to 12i+5j. And something you might want to do, maybe I will do it in a future video is actually draw out these two vectors and add them visually and you will see that you will get this exact answer. And as we go into further videos or future videos, you will see how this is super useful once we start doing more complicated physics problems or once we start doing physics with calculus. Anyway, I am about to run out of time on the 10 minutes so I will see you in the next video.