12.5: Equations of Lines & Planes (1/2)

okay so twelve five equations of lines and planes today we'll talk about liens and probably tomorrow we'll get to planes we will be writing the equations of lines in 3-space so to do that two pieces of information that we need we need a point a point that's on the line and we need the position by position you can also say the direction that the line is going so let's say we draw some line in 3-space say that is our line l let's say that we also know that we have this point on the line let's call that point P not so that point will be X not Y not Xena okay so then our figure we know that P naught is a point on the line so that's first thing we need we also said that we needed the direction for a line the direction will be determined by a parallel vector call it D so here's the idea if you know this point peanut pick another point on the line let's call it P which is X Y Z so ultimately we want to be able to find all the different P's all the different points on the line so if we were to draw vectors to these points this vector a then would be the direction vector it's more helpful though to have a vector that starts at the origin so what you're gonna do is you're gonna draw in some parallel vector so that vector V is parallel to a so we have this vector tells us the direction of the line and our point P naught tells us the physician so these are the two things that are necessary to write the equation of a line so let's say a then that a that I wrote in that's a vector that starts at P naught and head towards P if that's the case then another way it's write a is T multiplied by V they're parallel so they have the same direction their only differences their magnitude so that's what the T accounts for so this leads us to the vector equation of the line you will be learning three different forms of writing the line first one is a vector they're all related vector of equation of a line is going to be R equals R naught plus T times V this is the are not so R naught is the vector that goes to the point that we know R is this vector right here so when we plug in different values you have values of T it's gonna give us different points on the line plug in a value of T and it's gonna give you R so we'll give you this vector which will tell you a point that's on the line if you plug in values where T is greater than 0 your point is gonna be to the right of the initial point if you plug in values of T that are negative less than 0 you're gonna get points to the left of the initial point that makes sense ok another way that you're gonna see this written that vector R that gives us a point that's on the line it could be expressed in bracket notation X Y Z or not might be expressed as also vector notation or bracket notation the might be expressed as a b c so then instead of our you're going to have a X Y Z equals x naught plus T times a why not add T times B Z not add T multiplied by C ok now how does that help us that leads us to the next expression of a line which is the parametric equation of a line so the parametric equation for a line through some point X naught Y naught Z naught so that's exactly what we had above and parallel to a direction vector ABC I guess parametric equations it's going to be exactly what we wrote above so it'll be x equals X naught plus T times a y equals y naught add B times T or T times B and Z equals Z naught plus C times T so those are the first two I could ask you to write a vector equation I could ask you to write parametric equations questions on our equations before we do an example okay so example one find a vector and parametric equations for the line that passes through 0.42 and is parallel to the vector V which is negative 1/5 okay so first we're going to start with that vector equation the vector equation was R is equal to R naught add T times V our gun represents any point on the line so that's gonna stay R naught is the vector that goes from the origin to the point that we know so that'll be the vector for 2 + T V is any vector parallel to the line so in this case that vector that we got is negative 1 5 I am perfectly okay if you leave the equation like that you might seen it see it written like that sometimes you might see I and J notation so for I at 2 J plus T times negative I plus 5 J that's another way you might seen it rip see it written and then lastly some people like to put together the eyes and put together the J's so some people are gonna write it like this for I and then negative I so for minus TI plus 2 plus 5t j so any of those are acceptable and I believe WebAssign should allow you to put any of those in unless they specify otherwise so that's your vector equation we also need the parametric equations so X is going to equal four minus T Y is going to be two plus 5t and on those two obviously go together okay so this is some of what your homeworks gonna be so it looks long but a question like this should be pretty quick two questions for you my first question if I asked you to find three points on the line could you any three points on the line what would you do yeah plug in different values for T so if you plug in one you get the point three seven so that's one point on one other thing I wanted to ask you are these equations unique so is this the only set of parametric equations for this line is this the only vector equation let's think about the vector one is this the only vector equation for this line no why not there's other parallel vectors right couldn't I change the length of this vector and it's still parallel couldn't I choose a different point okay so not unique not unique okay next example we are going to find parametric equations of the line passing through point P 1 which is 2 4 negative 1 and point P 2 which is 5 0 7 okay so we talked about earlier we need a point and we need a pair a little better we obviously have a point there's two of them we have to choose from how do we find the parallel vector she used for the equation of our lines yeah exactly we're gonna find the vector connecting these two points so five minus two is gonna give us three zero subtract 4 is negative 4 7 subtract negative one is eight now one thing to note this is one parallel vector so any scalar multiple is gonna work so if you want to divide by a constant or multiply by a constant so it'll work we need parametric equations what point do you guys want to use the first one second one second great we'll use that one okay so we get x equals five plus 3t y equals zero minus 4t Z equals 7 plus 8 t so those are parametric equations questions on parametric or vector equations before we move on to the third representation okay last thing I want us to talk about with this problem before moving on is where does this line intersect the XY plane so I'm looking specifically for a point to the point where this line will intersect the XY plane ideas Z equals zero right XY plane will mean z equals zero so we get 7 plus 18 equals 0 t-then is negative 7/8 so you're gonna plug negative 7/8 into your parametric equations do you guys trust me if I just tell you what you get okay so the point ends up being 19 over 8 comma 7 over 2 comma 0 into these equations up here okay ready for the third representation of a line okay great it's called symmetric equations have you guys learned so much of equations for lines no that doesn't sound familiar okay okay this is based off of the parametric form so parametric form is X naught plus 80 y equals y naught plus BT Z equals Z naught plus CT for the symmetric equation what you're going to do or symmetric equations you're going to solve each of these parametric equations for T so the top one T will be X minus X naught over a for why it'll be T equals y minus y naught over B for Z it'll be T equals Z minus Z naught over C okay this then is the symmetric equation of a line X minus X naught over a equals y minus y naught over B equals Z minus Z naught over C okay so I'm gonna be honest with you I have no idea why you need to know three different equations for the same line but you're gonna be asked for all three not all three but you need to know all three in your homework you're gonna be asked for specific forms of the line and on the tester to look okay next example we are going to write symmetric equations for the line from number two so if you remember our vector V that we found will sit reading negative for 8 our initial point we used five zero seven so our equations then are going to be X minus five over three equals y minus zero so that's Y over negative four equals Z minus 7 over negative eight okay next thing we need to talk about when it comes to lines besides writing equations of lines related very closely to that is line segments so we are going to find parametric equations for the line segment joining P which is 2 for negative 1 and Q which is 5 0 7 ok so we're going to start by writing the equation of the line and then we'll figure out how to make it a line segment how do I new start my guys fine PQ okay did you guys notice these are at the exact suit teh same two points from before so that's the same exact vector I don't feel like we need to keep calculating vectors over and over again okay I'm going to use this point instead this time so we get x equals two plus 3t y equals four plus four T oh yes thank you and then positive eight I think okay and then Z equals negative one and eight T now this right now represents an entire line not just a segment from point to point so how would we make it a segment how would we get only part of a line rather than the whole line restriction on teeth okay so now we got to figure out what we need to restrict T to any ideas sure what is just a good guess for one value of t 0 plug in 0 see what you get if you plug in 0 you get 2 for negative 1 which gives us that one right there any guesses for what to plug in for this one plug in 1 get that point so T is gonna be between 0 & 1 so if it's a segment you have to make sure that you restrict T it's always gonna be between 0 & 1 just based on how we formed this line okay you might see another form though that I want you to see sometimes written like this R of T equals one minus T times R not plus T times R one so this is another one that you can use R naught is the vector from the origin to the first point R 1 is the vector from the origin to the second point this is also written sometimes as R naught plus T times R 1 minus R naught which is exactly what we just did over there okay we got one more equation of lines question and then we're gonna move on to Plains does that work for you guys okay so next example consider the following lines I want to know are the lines parallel or do they intersect so here are your two lines line one they're all going to be written parametrically line one is x equals one plus 40 y equals 5 minus 40 and z equals negative 1 plus 5t such first line second line x equals 2 add 80 y equals 4 minus 3 T and Z equals 5 plus okay so let's start with figuring out if the lines are parallel how would we know if the lines are parallel what the zero vector okay sure we could look at the cross-product what else can we look at because I'm gonna be honest I'm kind of lazy and I don't really feel like doing the cross-product so try to find the most efficient way to do this what two pieces of information do does a line give us the equation of a line point and directions so if the vectors are parallel what are we going to look at point or Direction direction and what should be true of the direction they have to be exactly the same scalar multiples of each other okay so I'm gonna start by rewriting line 1 I'm gonna write it in vector notation instead so choose your point 1 5 negative 1 plus T times 4 negative 4 5 this is just gonna help me to identify the direction a little bit easier so the point on the second line is 2 4 5 and then the direction is 8 negative 3 1 so these then are our two direction vectors not the same not scalar multiples so that tells us that our lines are not parallel okay next question up is do they intersect if they intersect where do they intersect no this one I think is the trickiest not to find it but just to understand okay do we recognize that these two T's are not the same T that the lines may intersect and if they do it wouldn't necessarily be at the same value of T do you get what I'm saying so T might be one in this case in two here and that's when they intersect so these T's we need to treat differently so to figure out if they intersect we're going to try to find their point of intersection if we look at the X's we get one plus four T one is equal to two plus eight T 2 this is line one line two and then for y we have five minus 4 T 1 equals 4 minus 3 T 2 and then Z negative 1 plus 5 T 1 equals 5 plus T 2 so this is always line 1 line 2 line 1 line 2 by one line to do this make sense why we have to use T 1 and T 2 okay sometimes you're gonna have one that's easier than the rest of them in this case like one that's easier to solve for I don't think there really is so I started with the first equation and I solve for T 1 if you trust me T 1 ends up being 1/4 plus 2t 2 right so this is a system so that what we're going to do is we are going to plug this into the second equation so we get 5 minus 4 instead of T 1 we're going to use 1/4 plus 2t 2 equals 4 minus 3 T 2 so this ends up being 5 minus 1 subtract 8 T 2 equals 4 minus 3 T 2 that's 4 that's 4 they cancel out you should end up seeing that T 2 equals 0 okay what do we do from here into all of them just one of them which one okay so let's try that plug T 2 into here if we plug T 2 in that'll be 0 so we get T 1 equals 1/4 okay did we need these two equations okay well let's try plugging these two into our two equations and see where they intersect right because from what we've found it looks like they intersect at the point where those those two are true with me okay so we're gonna plug 1/4 into line one oh this one right here we plug in 1/4 there we get two we plug in 1/4 there we at 4 and here we get 1/4 and you guys verify that I did that right okay line two we're gonna plug in 0 this time we get to 4 5 so do we need the third set of equations yes so what you need to do is before you even get to the point of plugging in you need to plug these two into the third equation to make sure that it's true I plug in 1/4 here like we did before we get 1/4 be plugging in 0 you get 5 what does that tell us about our two lines they do not intersect how is that possible that they're not parallel and that they don't intersect it's a throwback to geometry guys what do you call it if you have two lines in 3-space that do not intersect and are not parallel starts with an S four letter word oh so embarrassed for you all skew yeah skew lines skew lines that parallel don't intersect that sound familiar at all can I go off on a tangent for a minute before we talk about planes okay do we have any questions on lines before we go to my tangent and then back to planes okay what have you always been taught about parallel lines like an old map from the time that you learned about parallel lines what were you talking same slope and don't intersect ready not true that's not true at all those of you who take math classes in college the kind of math world that you all live in is one math world it's called normal Euclidean geometry it's like a normal piece of paper if you take a geometry class specifically in college you're going to learn about there's two other different kind of geometric worlds hyperbolic I don't remember the other one and in those worlds parallel lines sometimes intersect doesn't this low your mind can I myself draw it there are yeah there are computer programs I can draw it I certainly cannot though I felt like I was lied to though when I got to college we'll learn that no you don't feel like you were larger yeah yeah okay great now we're talking about planes planes are not too bad maybe I'll like a tiny bit more work sometimes but not that bad okay so unlike a line a vector that is parallel to a plane is not enough information to determine that plane so if I give you a point and I give you a vector parallel to a plane that is not enough information to write the equation of all the eight unique plane there are multiple planes that will be parallel to that vector and through that point so instead you need a vector that is perpendicular to the plane so that's the biggest difference you got an O with a line has to be parallel but the plane has to be perpendicular so here's gonna be our equation of the plane you are gonna take n which is called the normal vector that's a vector perpendicular to the plane you are gonna dot it with r subtract R naught so this is the normal vector R naught is a point on the plane if that vector is normal to the plane and we adopt these two what should we get zero so you're gonna get zero this is called the vector equation of a plane sometimes you'll be asked for the vector equation sometimes people are actually going to take the dot product of this so that normal vector let's say is the vector ABC remember that R represents any point on the line or a vector that goes to that point at least so R is XYZ so this will be X minus X naught y minus y naught Z minus Z naught so if we multiply this out it will lead us to a multiplied by X subtract X naught plus B times y minus y naught plus C times V minus Z naught equals 0 this is called the scalar equation of the plane and then there's one or other equation when all of this is multiplied out and you combine the like terms okay ready to use this idea for an example yeah great thank you for answering and not just staring at me this is example six we are going to find an equation of the plane passing through the point 3 negative 1 7 and perpendicular to vector n which is 4 2 negative 5 okay so like before this is not gonna be a unique equation that's just gonna be a one equation that's based on this point in this vector we could choose a different point we could choose a different vector so what we're gonna do is we're gonna take our normal vector n which is that for two negative five it's going to be dotted with r minus r naught so r again is XYZ she'll be XYZ - our point so X minus three comma y plus 1 z minus 7 equals 0 so this is the vector equation of a plane you can leave it like that other way you might express this is 4 times X minus 3 plus 2 times y plus 1 minus 5 times e minus 7 equals 0 that'll be the askew scalar equation of the plane so if I don't ask you for a specific set either of those are okay and then like I said the last way that people do it is they multiply out the 4 2 and negative 5 if you did that you would get 4x plus 2y minus 5z we have negative 12 and 2 so negative 10 positive 35 so it was positive 25 all three of those are okay this form is called the linear equation of a plane if you're asked for a linear equation it's that one now if you are asked to graph a plane the linear equation is most helpful so if I gave you this equation how would you go about graphing the equation of the plane what information is that equation give you yes you're gonna find the three different intercepts so plug in x and y or 0 find the Z intercept plug in X and Z or 0 find the y intercept etc can you guys picture that without us actually doing it ok sorry I know that our notes are long today next example we are gonna find an equation of the plane that passes through the following three points P is 1 3 2 Q is 3 negative 1 6 and R is 5 to 0 okay so if we've talked about a few minutes ago we need a point and we need a normal vector we have three points to choose from how are we going to find a normal vector so we have a plane with these three points how can I use those three points to find a vector that is perpendicular to the plane very good so we're gonna find two vectors I'm gonna do Max's suggestion and find PQ and PR then when I take their cross product I'll end up with a vector that's perpendicular to both PQ if I do 3 minus 1 I get 2 negative 1 minus 3 is negative 4 and 6 minus 2 is 4 and then PR 5 minus 1 is 4 2 minus 3 is negative 1 0 minus 2 is negative 2 I crossed them this is where I'm gonna have to do my determinants my ijk is it okay if I don't show all the work if we just okay great so eyes we're gonna ignore we get negative eight subtract four like that right so we get eight add four is gonna give us 12 for the J's we get negative four subtract 16 negative 20 but you have to flip sign is positive 20 negative to add 16 will give us 14 okay you can definitely use this as and if you want to any scalar multiple is still gonna be perpendicular to the plane so I normally simplify if I can so I divide it by 2 and got 6 10 7 so then our equation is going to be 6 use any of the points it doesn't matter I use to point P so 6 times X minus 1 plus 10 x y minus 3 plus 7 times Z minus 2 equals 0 that form is ok if you choose to multiply everything out you get 6x edy 10 Y add 7 Z equals 50 that's another possible okay questions about equations of planes okay we are not going to be able to get through the next example so tomorrow we have three more pages in it I think we got through enough today that it's not going to take us the entire period tomorrow so you should hopefully have time at the end of the period

Channel: Alexandra Niedden

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Length: 41min 21sec (2481 seconds)

Published: Mon Aug 26 2019