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Current time:0:00Total duration:24:46

Parametric representations of lines

Video transcript

everything we've been doing in linear algebra so far you might be thinking as it's kind of a more painful way of doing things you already knew how to do you've already dealt with vectors or I'm guessing that some of you all have already dealt with vectors in your calculus or your precalculus or your physics classes but in this video I hope to show you something that you're going to do in linear algebra that you've never done before and that it would've been very hard to do had you not I guess been exposed to these videos but I'm going to start with once again a different way of doing something you already know how to do so let me just define some vector here some vector instead of making them bold I'll just draw it with the arrow on top I'm going to define my vector to be I could do it with the arrow on top or I could just make it super bold I'm just going to define my vector it's going to be a vector in r2 let's just say that my vector is the vector to one if I were to draw it in standard position it looks like this you go two to the right and up one like that that's my right there that is my vector V now if I were to ask you what are all of the possible vectors I can create so let me define a set let me define a set s and it's equal to all of the vectors I can create if I were to multiply V times some constant so I multiply some constant some scalar times V times my vector V and I'll just to maybe a little bit formal I say such that C is a member I don't want to write C is a member of the real numbers C is a member of the real numbers now what would be a graphical representation of this set well if we draw them all in standard position I mean C could be any real number so if I were to multiply C could be to if you know if C is to C let me do it this way if I do 2 times our vector I'm going to get the vector 4 to let me draw that in standard position 4 - it's right there it's this vector right there that vector right there it's collinear with this first vector it's along the same line but just goes out to further now I could have done that I could have done 1.5 times our vector V let me do it in a different color and maybe that would be I'd be what that would be 1.5 times 2 which is 3 1.5 where would that vector get me I'd go 1.5 and then I'd go three and then 1.5 I'd get right there and I can multiply it by anything I could multiply 1 point 4 9 9 9 times vector V and get right over here I could do minus point oooo 1 times vector V let me write that down I could do 0.001 times our vector V and where would that put me it would put me right little super small vector right there if I did minus 0.01 and make a super small vector right there putting in that direction if I were to do minus 10 I would get a vector going in this direction that goes away like that or it goes way like that but you can imagine that if I were to plot all of the vectors in standard position at all of them that that could be represented by any C and real numbers I'll essentially get I'll you know I'll end up drawing a bunch of vectors where their arrows are all lined up along this line right there and all lined up in even the negative direction let me make sure I draw it properly along that line like that I think you get the idea so it's a set of collinear vectors so let me write that down it's a set of co-linear collinear vectors and if we view if we view these vectors as position vectors that you know that this vector represents a point in space in in r2 you know this r2 is just our Cartesian coordinate plane right here in every direction if we view this vector as a as a position vector let me write that down as a position vector if we view it as kind of a coordinate in r2 then this set if we visually represent it as a bunch of position vectors it'll be represented by this whole line over here it'll be represented by this whole line over here and I want to make that point clear because if I didn't if it's essentially a line of slope to right your rise sorry one half your rise is one your rise is one for going over to but I want I don't want to go back to our Algebra one notation too much but I want to make this point that this line of slope to that goes through the origin this is if we draw all of the vectors in this set as in their standard form or if we draw them all as position vectors if I didn't make that that clarification or that qualification I could have drawn these vectors anywhere right because I you know this to this for two vector I could have drawn I could have drawn over here and then I you know to say that it's collinear probably wouldn't have made as much visual sense to you but I think this Co linearity of it makes more sense to you if you say let's draw them all in in standard form all of them start at the origin and then they're there their tails are at the origin in their heads go essentially to the coordinate they represent that's what I mean by their position vectors they don't necessarily have to be position vectors but for I guess the visualization in this in this video what let's stick to that now I was only able to represent something that goes through the origin with this slope so you can almost view that this vector kind of represented its slope this was you almost want to view it as a slope vector if you wanted to tie it in to what you learned in algebra 1 what if we wanted to represent other lines that had that slope what if we wanted to what if we wanted to represent what the line that goes the same line or I guess a parallel line that goes through that point over there the point you know this is the point 2 comma 4 or if we're thinking in position vectors we could say that point is represented by the vector that point is represented by the vector let me that point is represented by the vector now we call that X it's represented by the vector X and the vector X is equal to 2 for that point right there what if I want to represent the line that's parallel to this that goes through that point 2 4 so I want to represent this line right here I want to represent draw it is parallel to this as I can I think you get the idea and it just keeps going like that in every direction these two lines are parallel how can I represent the set of all of these vectors or the drawn in standard form or the vecto love the vectors that if I were to draw them standard form would show this line well you could think about it this way if every one of the vectors that represented this line if I start with any vector that was on this line and I add my X vector to it and I add my X vector to it I'll be I'll show up at a corresponding point on this line that I want to be at right if I start if I take if I start and let's say that this is let's say I do negative let's say I do negative 2 times vector original so minus 2 times my vector V that equalled what minus 2 minus 4 minus 2 so that's that vector there but if I were to add X to it if I were to add my X vector if I were to add so if I were to do minus 2 times my vector V but I were to add X to it so plus X I'm adding this vector 2 comma 4 to it so from here I'd go right to and up 4 so I'd go here or visual you could just say heads to tails I would go right there so it'd end up at a corresponding point over there I'll end up at a corresponding point over there so the set of all points so when I define my set s as the set of all points where I just multiplied V times the scalar I got this thing that went through the origin but now let me define another set let me define a set L maybe L for line that's equal to the set of all vectors where it's X the vector X I can do it bold or I'll just do an arrow on it plus plus some scalar I could use even let me use T because I'm going to call this a pair of Metro ization of the line so plus some scalar T times my vector V times my vector V where or such that T could be any any member of the real numbers so what is this going to be this is going to be this blue line i if i if i were to draw all of these vectors in standard position i'm going to get my blue line for example if i do - - if this is -2 times my vector v i get here then if i add X I go there so the vector this vector right here that has its endpoint right there its endpoint sits on that line I could do that with anything if I go if I take this vector this is some scalar times my vector V and I add X to it I end up with this vector whose end point if I viewed as this position vector its endpoint dictates some coordinate in the XY plane so it'll it at that point so I can get to any of these vectors this is a set of vectors right here and all of these vectors are going to point they're essentially going to point to something when I draw them in standard form if I draw them in a standard form they're going to point to a point on that blue line now you might say hey Sal this was a really obtuse way of defining a line I mean we do it in algebra 1 where we just say hey you know Y is equal to MX plus B and we figure out the slope by figuring out the difference of two points and then we do a little substitution and this is stuff you learned in seventh or eighth grade this was really straightforward why am i defining this this obtuse set here and and and you know making you think in terms of sets and vectors and adding vectors and the reason is is because this is very general this is very general this worked well in r2 so in r2 this was great I mean we just have to worry about X's and Y's but what about the situation I mean notice in your algebra class your teacher never really gave you told you much or at least the ones I took about how to how do you represent lines in three dimensions and I mean maybe some classes go there but they definitely didn't tell you how do you represent lines in four dimensions or 100 dimensions and that's what this is going to do for us right here I defined X and V as as vectors in r2 there are two dimensional vectors but we can extend it to an arbitrary number of dimensions so let me just just to kind of hit the point home let's do one more example in r2 where it's kind of the classic algebra problem where you need to find the equation for the line or but here we're going to call it this the set definition for the line let's say we have two vectors let's say we have the vector a we have the vector a which I'll define is you know let me just say it's two one so if I were to draw in standard form it's two two one that's my vector a right there and let's say I have vector B let me define vector B I'm going to define it as I don't know let me define it as zero three so my vector B zero I don't move to the right at all and I go up so my vector B will look like that my vector B now I'm going to say that these are position vectors that they're at we draw them in standard form when you draw them in standard form the their end points represent some position so you can almost view these as coordinate points in r2 this is r2 i mean if i then all of these coordinate axes I draw are going to be r2 now what if I asked you give me a parametrizations the line that goes through these two points so essentially I want the equation you can if you're thinking in algebra one terms I want the equation for the line that goes through these two points that goes through those two points so the classic way you would have figured out the slope and all of that and then you just substitute it back and but instead what we can do is we can say hey look this line that goes through both of those points or you can almost say that you know that both of those vectors lie on I guess that's a better both of these vectors lie on this line now what is the what vector can be represented by that line or even better what lot what what vector if I take any arbitrary scalar multiple well this can can represent any other vector on that line and let me do it this way what if I were to take so this is vector B here what if I were to take B minus a we learned in I think it was the previous video that B minus a you'll get this vector right here you'll get the difference in the two vectors this is the vector B minus the vector a and you just think about it what do I have to add to a to get to B I have to add B minus a so if I can get the vector B minus a alright and we know how to do that we just subtract the vectors and then multiply it by any scalar then any scalar then we're going to get any point along that line we have to be careful so what happens if we take T so some scalar times our vector times the vectors B minus a B minus a what do we get then so B minus a looks like that but if we were to draw it in standard form remember this in standard form if B minus a would look something like this it would look something like this right it would start at 0 would be parallel to this and then from 0 we draw its endpoint so if we just multiplied some scalar times B minus a we would actually get we would actually just get rent we would get points or vectors that lie on this line vectors that lie on that line right there now that's not what we set out to do we wanted to figure out a an equation or a parametrizations if you will of this line or this set what's called the set L so we want to know what that set is equal to so in order to get there we have to start with this which is one of these which is this line which is this line here and we have to shift it and we could shift it either by shifting it straight up we could add vector B to it so we could take this this line right here and add vector B to it and so any point on here would have its corresponding point there so when you add vector B essentially shifts it up that would work so we could say we could add vector B add vector B to it and now all of these points for any arbitrary T is a member of the real numbers will lie on this Green Line or the other option we could have done is we could have added vector a vector a would have taken any arbitrary point here and shifted it that way right you would be adding vector a to it but either way you're going to get to the Green Line that we cared about so you could have also defined it as the set a vector a Plus this line essentially T times vector B minus a where T is a member of the reals now all of this so my definition of my line could be either of these things the definition of my line could be this set or it could be this set and all of this seems all very abstract but when you actually deal with the numbers it actually becomes very simple it becomes arguably simpler than what we did in algebra 1 so this L for these particular case of a and B let's figure it out my line is equal to let me just use the first example it's vector B so it's the vector 0 3 plus T times the vector B minus a well what's B minus a 0 minus 2 is minus 2 3 minus 1 is 2 4 t is a member of the reals now if this still seems kind of like a convoluted set definition for you I could write it in terms that you might recognize better if if we want to plot points if we call this the y-axis and we call this the x-axis and if we call this the x-coordinate or maybe more properly that the x-coordinate and call this the y-coordinate then we can set up a parrot we could set up an equation here this actually is the x with respect well I don't wanna this is the x-coordinate that's the y-coordinate or actually even better whatever actually let me be very careful there this is this is always going to revolt end up being coming some vector l1 l2 right this this is a set of vectors and any member of the set is going to look something like this so you know this could be this could be you know L I so this is the x coordinate and this is the y coordinate and just to get this chord in it and just to get this in a form that you recognize so we're saying that L is a set of you know this vector X plus T times this vector T times this vector B minus a here if we wanted to write it in kind of a parametric form we can say since this is what determines our x coordinate we would say that X is equal to 0 plus T times minus 2 or minus 2 times T and then we can say that Y since this is what determines our y coordinate y is equal to 3 plus T times 2 plus 2t so we could have rewritten that first equation is just X is equal to minus 2t + y is equal to 2t plus 3 so if you watch the videos on parametric equations this is just a traditional parametric definition of this line this line right there now you might have still viewed this as how this was a waste of time this was convoluted you had to define these sets and all that but now I'm going to show you something that you probably hit well unless you've done this before I guess that's true of anything but you probably haven't seen it in your traditional algebra class let's say I have two points and this now I'm going to deal in three dimensions so let's say I have one vector I'll just call it point one because these are position vectors or let's call it position one and it's the point this is in three dimensions let's make up some numbers negative 1 2 7 let's say I have point two and once again this is in three dimensions so you you have to specify three coordinates this could be the X the Y and the Z coordinate 0.2 I don't know let's look at 0 3 & 4 now what if I wanted to find the equation of the line that passes two through these two points in r3 so this is an r3 in r3 well I just said that the equation of this line so I'll just call that or the the set of this line let me just call this you know L it's going to be equal to we could just pick one of these guys it's going to be it could be p1 the vector p1 these are all vectors be careful here the vector p1 plus some random parameter T this T it could be time like you learn when you first learn parametric equations times the difference of the two vectors times p1 and it doesn't matter what order you take it so that's a nice thing to P 1 minus P 2 it could be P 2 minus P 1 and then this you know because this can take on any positive or negative value where T is a member of the real numbers so let's apply it to these numbers let's apply it right here what is p1 minus p2 p1 minus p2 is equal to let me get some space here p1 minus p2 is equal minus 1 minus 0 is minus 1 2 minus 3 is minus 1 7 minus 4 is 3 so that thing is that vector and so our line can be described as the set of vectors that if you were to plot it in standard position it would be this set of position vectors it would be p1 it would be we do that in green it would be minus 1 2 7 I could have put p2 there just as easily plus plus T times x minus 1 minus 1 3 where or such that T is a number of the real numbers now this also might not be satisfying for you you're like gee you know how do I plot this in three dimensions whereas my XY z-- and Z's and if you want to care about XY z-- and Z's let's say that this is let's say let me go down here let's say that this is the z axis this is the x axis and let's say the y axis it kind of goes into the into our board like this so the y axis comes out like that our y axis comes out like that so what you can view this and actually I probably won't grab so we the determinant for the x coordinate just our convention is going to be this term right here so we can write that X let me write that down so that term is going to determine our x coordinate so we can write that X is equal to minus 1 be careful what the color is minus 1 plus minus 1 times T plus minus 1 times T that's our x coordinate now our y coordinate is going to be determined by this part of our vector addition because these are the Y coordinates so we could say that our y coordinate is equal to I'll just write it like this 2 plus minus 1 times T minus 1 times T and then finally our Z coordinate is determined by that there and the T shows up because it's T times 3 or I could just put this T into all of this so the z coordinate is equal to 7 plus T times 3 or I could say plus 3t and just like that we have three parametric equations and you know when we did it in our - I did a parametric equation but we learned in algebra 1 you could just have a regular Y in terms of X you don't have to have a parametric equation but when you're dealing in r3 the only way to define a line is to have a parametric equation if you have just an equation with X and YS and Z's if I just have X plus y plus Z is equal to some number this is not a line not a line and we'll talk more about this in our 3 this is a plane this is a plane the only way to define a line or curve in three dimensions it's you know if I wanted to describe the path of a fly in three dimensions it has to be a parametric equation or if I were you know if I shoot a bullet in three dimensions and it goes in a straight line it has to be a parametric equation so DS I guess we call these are the equations of a line in three dimensions so hopefully you found that interesting and the I think this will be the first video where you have an appreciation that linear algebra can solve problems or address issues that you've never solved before I mean there's no reason why we had to just stop at three three coordinates right here we could have done this with 50 dimensions we could have defined a line in 50 dimensions or the set of vectors that define a line that you know two points sit on in 50 dimensions which is very hard to visualize but we can actually deal with it mathematically