Unit vector notation

good afternoon we've done a lot of work with with vectors and in a lot of the problems when we launched something into you know in the projectile motion problems or when you were doing the the inclined plane problems we always broke you know I always gave you a vector like I would draw a vector like this I would say you know something is has a velocity of 10 meters per second it's at a 30-degree angle and then I would break it up into the X&Y components so you know if I called this vector V I would use a notation V sub X and you know the V sub X would have been this vector right here V sub X would have been this vector down here the X component of the vector and then V sub y would have been the Y component of the vector and it would you know it would have been this vector so this was V sub X this is V sub y right and hopefully by now at second nature of how we would figure these things out V sub X would be 10 times cosine of this angle 10 cosine of 30 degrees which I think is square root of 3 over 2 but we're not worried about that right now and V sub y would be 10 times the sine of that angle this hopefully should be second nature to you if it's not you can just go through sohcahtoa and and say well the sine of 30 degrees is the opposite of the hypotenuse and you would get back to this but we've reviewed all of that and you should review the initial vector videos but what I want you to do now because this is useful for a simple projectile motion problem but once we start dealing with more complicated vectors and maybe we're dealing with multi dimensional vectors three dimensional vectors or we start doing linear algebra where we do n dimensional vectors we need a coherent way of an analytical way instead of having to always draw a picture of representing vectors so what we do is we use something that I call up and I think everyone calls unit vector notation so what does that mean so we define these unit vectors let me draw some axes it's important to keep in mind this might seem a little confusing at first but this is no different than what we've been doing in in our physics problem so far so let me draw the ax these draw the axes right there let's say that this this is this is 1 this is 0 this is X well sorry what am i doing this is 2 0 1 2 out of nose must have been writing in Arabic or something going backwards but it's a 0 1 2 that's not 220 and then let's say that this is 1 this is 2 in the Y Direction I'm going to define what I call the unit vectors in two dimensions so I'm going to first define a vector I will call this vector I and if this is the vector it just goes from it just goes straight in the X direction it has no y component and it has a magnitude of 1 and so this is I and we donate we denote the unit vector by putting this little cap on top of it there's multiple notation sometimes in a book you'll see this eye without the cap and it's just boldface there are some other notations but if you see I and then not in the imaginary number sense you should realize that that's the unit vector it has magnitude 1 and it's completely in the X direction and I'm 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'm dealing with two dimensions and as later we'll see in three dimensions it will actually be a third dimension and we'll call that K but don't worry about that right now but if we're dealing in in two dimensions we can we can define any vector in terms of some sum sum of these two vectors so how does that work well what this vector here let's 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 this and that's 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 the resulting sum is kind of where you end up right so would be if you added this vector and then you put this tail to this head and you end up there so you end up there so that's the vector so can we define V sub X as some multiple of I of this of this unit vector well sure V sub X completely goes in the X direction right but it's it doesn't have a magnitude of 1 it has a magnitude of 10 cosine of 30 degrees so its magnitude is 10 so this is let me draw the unit vector up here this is the unit vector I it's going to look something like this and this so V sub X it's in the exact same direction and it's just a scaled version of this unit vector and what's its what multiple is it of that of that unit vector well the unit vector has a magnitude of 1 this has a magnitude of 10 cosine of 30 degrees so I think that's what like five square roots of three or something like that so we can write V sub X as we could write V sub X I keep switching colors to keep things interesting we can write V sub X is equal to ten cosine of 30 degrees times that's the degrees times the unit vector I let me stay in that color so you don't get confused times the unit vector I does that make sense well the unit vector I goes in the exact same direction but the the X component of this vector is just a lot longer it's 10 cosine 30 degrees long and I can you know that's equal to cosine of 30 degrees is square root of 3 over 2 so that's five square roots of three I similarly we can write we can write the Y component of this vector is some multiple of J is some multiple of J so we could say V sub y the y component well what is sine of 30 degrees sine of 30 degrees is 1/2 so 1/2 times 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 this vector J of the unit vector J and what multiple is it well it has it has length 5 while the unit vector has just length 1 so it's just 5 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 in its Y component and we also know so this is a hole vector V what's its X component well its X component can be written as a multiple of the X unit vector that's that right there so you can write it as 5 square roots of 3 I plus its Y component so what's its Y component well its Y component is just a multiple of the Y unit vector which is called J with the little funny hat on top and that's just this it's 5 times J so we've done now by defining these unit vectors and I can switch this color just so you remember this is I that this unit vector is this is 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 have if I can write a vector in this format I can add them and subtract them without having to resort to resort to visual means and what do I mean by that let me so if I have you know if I define some vector a is equal to I don't know 2i plus 3j and I have some other event this is a vector and I have some other vector B this little arrow just means it's a vector sometimes you'll see it as a whole arrow sum as Oh ten-eye plus I don't know 2j if I were to say what's what's what's the sum of these two vectors a plus B before we had this unit vector notation we would have to draw them and put them heads to tails and it didn't you have to do it visually new 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's just 2 plus 10 times I plus 3 plus 2 times J and that's equal to 12 I plus 5 J and something you might want to do a maybe I'll do it in a future video is actually draw out these two vectors and add them visually and you'll see that you get this exact answer and as we go into further videos or future videos you'll see how this is super useful once we start doing more complicated physics problems or once we start doing physics with calculus anyway I'm about to run out of time for on the 10 minutes so I'll see you in the next video