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Current time:0:00Total duration:25:33

Video transcript

in the last videos a little formal and defining what what RN is and what a vector is and and and what vector addition or scalar multiplication is in this video I want to kind of go back to basics it just give you a lot of examples and give you a more tangible sense for what vectors are and how we operate with them so let me define a couple of vectors here so let me and I'm going to do most of my vectors I'm going to do in this video are going to be an r2 and that's because they're easy to draw are to remember r2 is the set of all two tuples ordered ordered two tuples where each of the number so you know you can have X 1 X 1 and let me see the my 1 looks like a comma X 1 and X 2 where each of these are real numbers so each of them X 1 is a member of of the reals X 1 and X 2 is a member of the reals and just to make you give you a sense of what that means if this is right here is my my coordinate axes and I wanted to plot all my X ones X 2's right this could be you know you could view this as the first coordinate we always imagine that as our x axis and then our second coordinate we plotted on the vertical axis that traditionally is our y-axis but we'll just call that the second number axis whatever you could visually represent all of our two by literally every single point on this plane if we were to continue off in infinity in every direction that's what our two is our 1 would just be points just along one of these number lines that would be our 1 so you could immediately see that our two is kind of a bigger a bigger space but anyway I said that I wouldn't be too abstract I would be I would show you examples so let's let's get some vectors going in r2 so let me define my vector a I'll make it nice and bold my vector a is equal to make some numbers up negative 1 2 and my vector B V make it nice and bold let me make that is I don't know three three one those are my two vectors now let's just let's just add them up and see what we get just based on my definition of vector vector addition and I'll just stay in one color for now so I don't have to keep switching back and forth so a nice deep a-plus bolded B is equal to I just add up each of those terms negative one plus three and then two plus one that was my definition of vector addition so that is going to be equal to two two and three fair enough this just came out of my definition of vector addition but how can we represent these this vector so we already know that if we have coordinates you know if I have the coordinate and this is just a convention it's just the way that we do it the way we visualize things if I wanted to plot the point 1 comma 1 I go to my coordinate axes the first point I go along the horizontal what we traditionally call our x-axis and I go one in that direction and then the convection is the second point I go one in the vertical direction so the point 1 1 I would sorry let me be very clear this is 2 & 2 so 1 is right here and 1 is right there so the point 1 1 would be right there that's just the standard convention now our convention for representing vectors are you might be simply 0 maybe I just represent this vector at the point minus 1/2 and on some level you can do that and I'll show you a second but the convention for vectors is that you can start at any point any point let's say this where we're dealing with 2-dimensional vector so you can start at any point in r2 so let's say that you're starting at the point so let's say that you start at the point x1 and x2 this could be any point any point in r2 in r2 to represent this vector what we do is we draw a line from that point to the point x1 and let me call this let's say that we wanted to draw a so x1 minus 1 so this is I'm representing a so this is I want to represent the vector a x1 minus 1 and then x1 plus 2 now that that if that seems confusing to you when I draw it will be very obvious so let's say I just want to start at the point let's just say for you know for quirky reasons I just pick a random point here I just pick a point that one right there that's my starting point so minus 4 comma 4 that's minus 4 comma 4 now if I want to represent my vector a what what I just said is is that I add I add the first term in vector a to my first coordinate so X 1 plus minus 1 or X 1 minus 1 so my new one is going to be so this is my X 1 minus 4 so now it's going to be let's see I'm starting at the point minus 4 comma 4 if I want to represent a what I do is I draw an arrow 2 minus 4 plus this first term minus 1 and then 4 plus the second term 4 plus 2 and so this is what this is minus 5 comma 6 so I go to minus 5 comma 6 so I go to that point right there and I just draw a line so my vector will look like this I draw a line from there to there and I draw an arrow at the end point so that's one representation of the vector minus 1/2 actually let me do it a little bit better because minus 5 is actually more a little closer right here minus 5 comma 6 is right there so I draw my vector like that like that but remember this point minus 4 comma 4 was an arbitrary place just to draw my vector I could have started I could have started at this point here I could have started the point 4 comma 6 and done the same thing I could have gone minus 1 in the horizontal direction that's my movement in the horizontal direction and then plus 2 in the vertical direction so I could have drawn so minus 1 in the horizontal and then plus 2 in the vertical gets me right there so I could have just as easily drawn my vector like that these are both interpretations of the same vector a and I should draw them the color of vector a so vector a was this light blue color right there so this is vector a this is vector a sometimes it'll be a little arrow notation over the vector but either of those reckon I can draw an infinite number of vector a is I could draw a vector a here I could draw it like that and vector a it goes back one back one and up two so vector a could be right there vector a similarly vector B what does vector B do I could pick some arbitrary point for vector B some zartra point it goes to the right three so it goes to the right one two three and then it goes up one so vector B one representation of vector B u looks like this another representative I could start could do it right here I could start it right here I could go to the right 3 1 2 3 and then up 1 this would be another representation of my vector of my vector B there's an infinite number of representations of them but the convention is to often put them in what's called the standard position and that's to start them off at 0 0 so your initial point so let me write this down standard position standard position is just to start the vectors at 0 0 and then draw them so vector a in standard position I'd start at 0 0 like that and I would go from I would go back 1 and then up 2 so this is vector a in standard position right there and then vector B instead let me write that that's a and then vector B in standard position is 3 go to 3 right and then up 1 but these are the vectors in standard position but any of these other things we drew or just as valid just as valid now let's see if we can get an interpretation of what happened when we added a plus B a plus B a plus B well if I draw that vector in standard position I just calculated now it goes to 3 so I go to the right 2 and I go up 3 so if I just draw it in standard position it looks like the it looks like this this vector right there that vector right there and at first when you look at it so this this vector right here is the vector a plus B in standard position when you draw it like that it's not clear what the relationship is when we added a and B but to see the relationship what you do is you put a and B head details and what does that means you put the tail end of B to the front end of a so if can remember all of these are valid representations of B all of the representations of the vector B they all have you can all their all parallel to each other but they can start from anywhere so another equally valid representation of vector B is to start at this point right here kind of the end point of vector a in standard position and then draw a vector B starting from there so you go three to the right so you go one one two three and then you go up one so then you go up one so vector B could also be drawn just like that just like that and then you should see something interesting it happened and remember this is this vector B representation is not in standard position but it's just an equally valid way to represent my vector now what do you see when i add a which is right here to be 2b what what do i get if i connect the starting point of a with the endpoint of b i get the addition i have added I have added the two vectors and I could have done that anywhere I could have started with a here and then I could have done the endpoint of I could have done started B here and gone 3 to the right 1 2 3 and then up 1 and I could have drawn be right there like that and then if I were to add a plus B I go to the starting point of a and then the endpoint of B and I that should also be the visual representation of a plus B and just to make sure it confirms with this number what I did here as I went to to the right 1 2 and then I went 3 up 1 2 3 and I got a plus B now let's think about what happens when we scale our vectors when we multiply it by us but times some some scalar factor so let me pick new vectors those have got a monopoly monotonous let me define vector V V for vector let's say that it is equal to 1 2 so if I just wanted to draw a vector V in standard position I would just go one to the horizontal and then 2 to the vertical that's it that's the vector in standard position if I wanted to do it in a non standard position I could do it right here one to the right up to look just like that equally valid way of drawing vector V equally valid way of doing it now what happens if I multiply vector V what if I have I don't know what if I have 2 2 times V 2 times my vector V is now going to be equal to 2 times each of these terms so it's going to be 2 times 1 which is 2 and then 2 times 2 which is 4 now what does 2 times vector V look like well let me just start from an arbitrary position let me just start right over here so I'm gonna go 2 to the right 1 2 and I go up 4 1 2 3 4 so this is what 2 times vector V looks like this is 2 times my vector V and if you look at it it's pointing in the exact same direction but now it's twice as long and that makes sense because we scaled it by a factor of 2 when you multiply it by a scalar you're not changing its direction its direction is the exact same thing as it was before you're just scaling it by that amount and I could draw this anywhere I could have drawn it right here I could have drawn 2 V right on top of V and then you would have seen it and I don't want to cover it you would have seen that it goes it's exactly in this case when I drew it in standard position its collinear it's along the same line it's just twice as far it's just twice as long but they have the exact same direction now what happens if I were to multiply if I were to multiply - let me say minus 4 times our vector V minus four times our vector V well then that will be equal to minus 4 times 1 which is minus 4 and then minus 4 times 2 which is minus 8 so this is my new vector minus 4 minus say this is minus 4 times our vector V so let's just start at some arbitrary point now let's just do it in standard position so you go to the right 4 or so you go to the left 4 so 1 so you go to left 4 1 2 3 4 and then down 8 look like that so this new vector is going to go and this is going to look like this I'll make sure I can draw a relatively straight line there you go so this is minus 4 times our vector V I'll do little arrow on it to make sure you know it's a vector now what happened well we're kind of in the same direction actually we're in the exact opposite direction but we're still along the same line right we're just in the exact opposite direction and it's this negative it's the negative right there that flipped us around if we just multiplied negative 1 times this we would have just flipped around 2 right there right when we multiplied it by negative 4 so we scale it by 4 so you make it four times as long and then it's negative so that it flips around it flips backwards so we can now that we have that notion we can kind of we can start understanding the idea of subtracting subtracting vectors let me write make up to two new vectors right now let's say my vector let's say my vector X nice and bold X is equal to and I'm doing everything in r2 but I'll do it in the last part of this video I'll make a few examples in r3 or r4 let's say my vector X is equal to is equal to 2 4 and let's say I have a vector Y why make it nice and bold and then that is equal to that is equal to negative 1 minus 2 and I want to think about the eye the notion of what X minus y is equal to X - why well we can say that this is the same thing as X plus minus one times our vector Y right so X plus minus one times our vector Y now we can use our definitions we know how to multiply by a scalar so we say that this is equal to let me switch colors I don't like this color this is equal to our X vector is 2 4 and then what's -1 times y so minus 1 times y is minus 1 times minus 1 is 1 and then minus 1 times minus 2 is 2 so X minus y is going to be these two vectors added to each other right I'm just adding the minus of Y this is this right here is minus vector Y so this X minus y is going to be equal to 3 and 3 and 6 so let's see what that looks like when we visually represent them our vector X was 2 4 so 2 4 in standard position it looks like this standard position it looks like this that's my vector X and then vector Y in standard position let me do it in a different color I'll do Y in green vector Y is minus 1 minus 2 so minus 1 minus 1 minus 2 so minus 1 minus 2 it looks just like this minus 1 minus 2 it looks just like this and actually ended up inadvertently doing collinear vectors but hey this is interesting too so this is vector Y so then what's their difference it says 3 6 so it's the vector 3 6 so it's this vector let me draw it someplace else so I start here I go 1 2 3 and then I go up 6 so then up 6 it's a vector that looks like this that's the difference between the two vectors so you say at first you say this is X minus y you're like hey how does this how is this the difference of these two well if you if you overlay this if you just shift this and over this you could actually just start here and go straight up and see that it's really the difference between the endpoints you're kind of connecting the endpoints but what I actually didn't want to draw cold in your vector so let me do another example although that one is kind of interesting you often don't see that one in a book let me define vector X in this case to be to be 2/3 and let me define vector Y to be minus minus 4 minus minus 2 so what would be X in standard position it would be 2/3 it looked like that that is our vector X if we start at the origin so this is X and then what does vector Y I look like I'll do Y an orange minus 4 minus 2 so vector Y looks like this vector Y will look like this now what is X minus y X minus y well you know we could view this 2 plus minus 1 times this we could just say 2 minus minus 4 I think you get the idea now but we just did it the first way the last time because I wanted to go from my basic definitions of scalar multiplication so X minus y is just going to be equal to 2 plus minus 1 times minus 4 or 2 minus minus 4 that's the same thing as 2 plus 4 so it's 6 and there's 3 minus minus 2 so it's 5 right so the the difference between the two is the vector 6 5 so you could you could draw it out here again so you could go to add 6 to 4 you go up there and then to 5 you'd go like that so the vector would look something like this to curve like that so that's X minus y but if we drew them between like in the last example I showed that you could draw it between their two heads so if you do it here what does it look like well if you start at this point right there you start at that point right there and you go 6 to the right so you go 6 to the right and then up 5 you end up right there so the difference between the two vectors let me make sure I get the difference between the two vectors looks like that it looks just like that which kind of should make sense intuitively X minus y that's the difference between the two vectors - you can view the difference is how do you get from one vector to another vector right like if you know let's get and go back to our kind of our second grade world of just scalars if I say what seven minus five is and you say it's equal to two well that just tells you that five plus two is equal to seven or the difference between five and seven is two and here you're saying look the difference between x and y is this vector right there it's equal to that vector right there or you could say look if I take five and add two I get seven or you could say look if I add if I take vector Y and I add vector X minus y then I get vector X now let's do something else that's interesting let's do what Y minus X is equal to Y minus X what is that equal to do it another color right here well we'll take minus 4 minus 2 which is minus 6 and then you have minus 2 minus 3 it's minus 5 so Y minus X is going to be let's see because if we start here we're going to go down 6 1 2 3 4 5 6 and then back 5 so back to 4 5 so Y minus X looks like this Y minus X looks like that it's really the exact same vector remember it doesn't matter where we start it's just pointing in the opposite direction so if we shifted it here I could draw it right on top of this it would be the exact as X minus y but just in the opposite direction which is this a general good thing to know so you can kind of view them as the negatives of each other and actually let me make that point very clear you know we drew why why I drew before actually let me draw X X we could draw is 2 3 so you go to the right 2 and then up 3 I've done this before this is X a non-standard position that's X as well what is negative x negative X is minus 2 minus 3 so if I were to start here I'd go to minus 2 and then I go minus 3 so minus X would look just like this minus X so it looks just like X it's parallel it has the same magnitude it's just pointing in the exact opposite direction and this is just a good thing to kind of really get seared into your brain is to have an intuition for these things now just I kind of finish up just to finish up this this kind of the idea of adding and subtracting vectors let me just do everything I did so far was in an r2 but I want to show you that we can generalize them and we can even generalize them to two vector spaces that aren't aren't normally intuitive for us to to actually visualize so let me define a couple of vectors let me define vector let me define vector a to be equal to 0 minus 1 2 & 3 let me define vector B to be equal to 4 minus 2 0 5 we can do the same addition and subtraction operations with them it's just it'll be hard to visualize but we can keep them in just vector form and and so we won't so that they're still useful to think in 4 dimensions so if I were to say 4 times a 4 times a this is the vector a minus 2 times B what is this going to be equal to and this is a vector what is this going to be equal to well we could rewrite this as 4 times this whole column vector 0 minus 1 2 & 3 minus 2 times B minus 2 times 4 minus 2 0 5 and what is this going to be equal to this term right here 4 times this you're going to get you're going to get actually the pen tablet seems to not work well there so I'm going to do it right here 4 times this you're going to get 4 times 0 0 minus 4 8 4 times 2 is 8 4 times 3 is 12 and then - I'll do it in yellow - 2 times 4 is 8 2 times minus 2 is minus 4 2 times 0 is 0 2 times 5 is 10 now this isn't a good part of my board so let me just it doesn't right well right over there I haven't figured out the problem but if I were to just write it over here what do we get what do we get with 0 minus 8 minus 8 minus 4 minus 4 minus negative 4 so that's minus 4 plus 4 so that's 0 8 minus 0 is 8 12 minus what was this I can't even read it what it says this was a 10 no now you can see it again something it's very bizarre 2 times 5 is 10 so it's 12 minus 10 so it's 2 so when we take this vector and multiply it by 4 and subtract 2 times this vector we just get this vector and even though even though you can't represent this in kind of an easy kind of graphic graphical format this is a useful concept we're going to see this later when we apply the sum of these vectors to multi-dimensional spaces