[Music] [Applause] okay in this video we're gonna have a look at vector proofs now this is quite a tough topic I'm gonna start this off on not the hardest vector proofs but bear in mind it is a tough topic anyway grab a piece of paper grab a pen try and draw these diagrams down as we're going over them have a go at working through some of them you might have to walk them you know rewind have another go have a little practice see if you can sort of replicate similar ones that I'm doing before we have a practice but we're gonna get started I'm gonna go over to and I've got two feet up I go up so hopefully so I'm gonna take us too long for this but it is you know up there in the top sort of grade eight nine topics so let's have a look we've got P is a point on a B such that a e 2 P P 2 B is 3 to 1 so obviously having a look at the diagram there AE to be we're looking at that line it says a to P and P to B is in the ratio 3 to 1 now the best thing we can do is turn this into fractions so we think about that that's a total of 4 parts so that's 3/4 and 1/4 so this part of the line here is 3/4 of the line and this part of the line here is 1/4 of the line and we always have to be able to try and move up and down part of this line we're gonna have a look at how to do that so it says find the vector o - P in terms of a and B and give your answer in its simplest form unless it's got a nice one it's what I'm starting off hopefully quite relatively easy on this one but a and B is already on the diagram there now if it wasn't we'd have to have we probably given these vectors we'd have to label them on ourselves but this has been put on for us so find oder P so trying to find out how to get from o here up to P and I normally like to draw a little line in for that but I'm not going to do it for this one okay I'm going to think how do I get from ode to P now I can't move in this sort of direction okay at the moment all I can do is I can move in terms of a and I can move in terms of B so I can either go along the bottom here and then try and find a vector to move upwards this way that's one option or let's get rid of that we could move up be this way and then try and go down to P this way and there's matter which one we choose we're just going to think about how we're going to do that now in order to do that I need to be able to move and down the line a V now just personal preference here I'm gonna have a look at how can I move up from a to P okay I could do it the other way around but I'm just gonna stick with doing it one way again whichever way we choose to do it that's fine but I'm just gonna have a look at this vector here I've made a piece I'm gonna write that down over here and that's the victim gonna try and find a tepee that's right down here a tepee that's gonna allow me to move up and down part that line so in order to get from a to P the first vector that I'm gonna have a look at is the full length of the line how do I get from a all the way up to B again if I can figure that out then I just need to be able to 3/4 of that from a to P and we can do that quite simply once we've worked it out but I've got to find a to be first so the first vector that I'm gonna actually find is A to B and when it comes to these vector questions you're always looking at how you can move up and down the line without any vectors on so that line A to B doesn't have any vectors on and that's all I'm we're gonna have to find now in order to do that I could to get from A to B I can go backwards this way so the reverse of a which would be minus a so it's write that in update I'd do - a map move me back down a and then I can go up 3b through positive B so B minus a plus B okay now at this point it's completely up to you you can actually rewrite this in a different way just to a remove having so many symbols I can swap them over I can have positive B minus a and just have it as B minus a ok and I do tend to swap them around quite often to avoid having the symbols but they mean exactly the same thing minus a plus B is the same as B minus a so there's that little vector that allows me to get from A to B that's an important one without that vector we can't actually answer the rest of the question so that's a very important one for us to look for how to move up and down the unknown line now I don't want to go for me to be I want to go just part of the line 3/4 of the line from a to P so what I can do is I can multiply this vector by 3/4 and I'm going to use that green one there that I've underlined the B minus a so to get from a into P yes we go do you have made two P we need to do 3/4 of that so I can open it up in a bracket like this just to make sure our times them both and I get 3/4 of B minus a now I am going to expand that bracket again I could leave it like that but it wants the vector o to P so one of the other answers a bit add some bits into this but if we just expand that out we get 3/4 B minus 3/4 of a and that allows me to get from A to B now that's a very important vector there that now allows me to move part way up the line and obviously the question asked me to get from o to P so we already know how to get from ODEs away that's a long here so to get from oats away and let's just start to write this down and let's have a think so from o to P let's just write it down in one step we would have to do the vector we'd have to go from o to a and then add to that this vector that we've got a to P and in total that gets me from o to P now I'll just take note of this is that I could have gone from o to B as well why it gets me up towards P but then if I wanted to get from o to be and I wanted to go up this direction I would have had to have worked out the vector B to P in order to move down this way ok so I chose not to do that at the start shows to work out the vector a to P that I've got in Green's point and it puts there so I can't move up this direction but it's ok cuz we've already got a to P so what we have to do is go along that a there so to get the motorway so I think about this to get from o to a it's the vector a so if I write all this down I'm going to get rid of this little bit in purple here so I can start writing it all down to get from o to a it's the vector a and go label this now o to P that would equal a which gets this remote away add this vector from above here which is a tepee and add 3/4 B minus 3/4 a ok and if you have a look here now we can actually join some of these up because we've got an A here or one a and minus a 3/4 a now if this sometimes help with some of the harder questions to think of how many quarters that a is just to keep it all matched and 1 a is 4 quarters a ok I might have to think about it four quarters a that's just the same as 1 4 divided by 4 is 1 so 4 quarters take away 3 quarters would leave me with 1/4 a so I simplify that much cautery yeah we go and then we still got the 3/4 B that's remaining unchanged so 3/4 B so we have 1/4 a at 3/4 B now a lot of the time and this is a key little bit here with them with vectors we normally have to factorize these vectors here little resultant vectors it's just the result of adding all these together and normally we need to factorize this and is a massively key part of this topic so I'm going to do su disappear I'm going to write factorize and it's very often that you can't factorize a whole number up you have to factorize a fraction augh if you have a look they both divided by 1/4 so if I wanted to factorize 1/4 out and that would be 1/4 the 1/4 a divides by 1/4 once I've been wanting and the 3/4 be three 1/4 fits into that three times BAE plus 3b and that would be our final answer there 1/4 a plus 3b ok and there we go and that is our vector to get from o to P in its simplest form not gonna have a look if something is a little little trickier here where we've got it proves that piece is a straight line so that was just like a little intro into vectors really but the key that there is moving up and down the line that doesn't have any vectors on and being able to do 3/4 or 1 quart or 1/2 of that I'm just multiplying out and being comfortable with writing these no vectors as they're written but I saw another one okay so looking at this question it says B is the midpoint of AC that's this one here so it says that this is a midpoint now straight away if you have a look on that line like a 2 B is the vector a and if that's the midpoint of B to C then this here on the other side has to be exactly the same so we can label that a straight away that's another way it says M is the midpoint of PB let's find that that's here M is the midpoint of PB and if it's the midpoint it's not in a ratio this time but a midpoint means it's in the ratio 1 to 1 or 1/2 and 1/2 here and it's also taking note of that lying there B P you can see doesn't have any vectors on every other line has a vector on but B 2 P or P to be whichever way we look at it has no vector on it so that line there is going to be one that we're gonna have to find how to move up and down it says show that NMC is a straight line so if I draw line let's have a look MC quite a party without a ruler but n MC there we go that's the line there which one to prove is a straight line now when it comes to these types of questions where you got to show it's a straight line we just need to have a look at this green line here so there's three vectors I can have a look at on this green line I can go right from the start from n all the way up to see that's a vector I can have a look at into C or I could do part of the line I could go from n just M that won't get me part way along the line so I got into in or I could go from M that line there with the cross on up to see there are three three possible vectors there that we could have a look at now when we're trying to prove something is a straight line I only have to find two of these and it tends to be that this one here is the easiest to find the full length of the line and I always look for that one first they're not just pick and choose between one of the other two and you're gonna see we're gonna be out of proofs and make such a straight line because there's going to be something that is similar within the vectors there when we actually find them okay what we'll go by actually just trying to work them out first and then we'll see at the end how it proves it's a straight line but n to C that's quite a nice easy one actually because I can go from in here I can go backwards up this to be I'm gonna write this down in the vector n to C so I can go - to be like - TV because I'm going backwards through that TV and then I can move all the way down the line here from A to C and that's two of those days there so plus 2ei there we go and there's the victor in to see I'm going to rub these lines out let's get rid of those okay so that's my back to there and again I can rewrite that if I want to get rid of the symbols I can write to a - TV okay so now we have the vector to a minus B that is from n to C now that's one of our vectors done so let's tick that off one of the three next let's have a look at one of the other ones and doesn't really matter which one we pick n - M or n m to see it doesn't really matter I just cover the second one is this a second one that I wrote and we just need to think how we're gonna get from n to M there okay so we have two options either we could go down here and then up there or we could go from n to a a to b and b down to m all the way around now that's three parts s I'm just gonna I'm just gonna ignore that one I'm just gonna go down and up okay so to get from n to P that's okay that's just down this little bit to be here but then I have to work out how to get from P to M now in order to get from Peter M okay this part here I need to be able to move up the full length of the line so I need to know what the full vector there for that line is and that line is the vector P up to B so I need to work out P to B now so let's not look at how would you go about getting P to be okay and then we'll think about adding all these up together in a sec so to get from P to B let's write this one down all right now don't over here P to be I have to go up here from 2a and then from a over here to B so to get from P to a and I saw that we have to go backwards through this B and backwards through another to B's so that's gonna be minus three B's so minus 3b and then Plus this a up here so plus a so minus 3b plus say okay now that gets me from P to B now we don't want to know Peter V do we want to know how to get from P to M which is gonna be half of that okay so to get from P to M I'm gonna have to have that okay because n is the midpoint when you want to go halfway up that line half of that vector so we just need to do what we did before we'll stick a half outside the bracket I'm not gonna rearrange it here to make it a minus 3b it's gonna do it as we as we've done it here so minus 3b plus a we're gonna do half of that okay so if we expand that I saw the look a half times minus three remember you can always think of these as fractions over one so 1/2 times minus 3 is minus 3 over 2 so minus 3 over 2 B plus and a half times 1 a is 1/2 a so plus 1/2 a there we go and that's our vector P to em now we're almost done because if we can get from n to P which we can we just go down our B now we can get from P to n using this vector here minus 3b plus Ave so if we have a look at this one here let's label this up so we're now going from n to M the full the full bit there and in order to do that we do 1b so be and then up to that this vector here that we've just worked out in case we're gonna bring that down here okay literally the one with just works out so be and I'm gonna put the the half a next to it let's have a look impact no let's put the minus three so it's not going to be plus because it's minus three and a half B so we could put add 2-3 and a half B that was going to get rid of the plus and just put it in there just to avoid having extra symbols here so it's B minus three and a half B plus a half a there we go right so again we just need to simplify this down we're almost done we need to simplify this and if we have a think let's have I think the bees there in terms of halves so one B is two halfs B and two halfs be take away three halfs being is going to leave us with minus 1/2 B okay one take away one and a half is minus 1/2 so if we write work that out we've got minus 1/2 B plus 1/2 1/2 a that we've got there and that's our final vector there to match that up now that means we've done in to him so we can tick that off okay now we've got two we just need to have a look and see if we can compare these now I mentioned in the last question it's very very important here that we factorize these vectors and we can see if there is anything common between them we'll have a look so if I come back to this into c1 the first one if I factorize this one here and rather than rearranging it I'll factorize this one I could take two out and if I take two out that would leave me with - a I'm sorry minus B plus a inside the bracket okay and that there is gonna be a really key vector so I'm going to highlight that - lots of minus B plus a if I do the same again to this one so if we factorize this one as well just see that was that one factorized and we factorize this one the only thing we can factorize now it's a half so I could take 1/2 out it's a look so take out a half divide the first bit by 1/2 we get minus B 1/2 divided by 1/2 is 1 so minus 1 D and then dividing a by 1/2 we get plus a there we go and that's that vector therefore if we factorize it and again I'm gonna highlight that one and let's just write them down over here next to these vectors that I was going to work out in the first place so for n to see we had we have to lots of minus B plus a and for n to M the other one we worked out we've got 1/2 brackets minus B plus a there we go so if you have a look we've obviously got this similar racket we've got this same vector in there minus B plus a now if we've got the same vector there minus B plus a it means that the lines are going in the same direction and if they're going in the same direction and they're connected here at em then they must be on the same straight line because if they go in same direction they're connected together at that point M then they have to be going in the same straight line the two they're in m2 C means that's the full length of that so it's doing to lots of that vector the half means that from n to M is half of that vector which means if we had have worked out this line here m2 see it would have been through the other three halves or the other one and a half of that vector okay so you get three halves of minus B plus a and you could work that one out as well but we only ever have to work two out now that is enough to show that it's a straight line but personally a particularly if it ever says prove or anything like that I just like to write that they both share the same vector you can say that they share different multiples of the vector ones too lots of it ones half lots of it but you would I would normally put a little statement in just pointing an arrow at those brackets and saying they share the same vector okay so obviously that is quite a tricky one it's not not some of the hardest of the vectors some of these new vectors seem to be very very tough but there's a lot going on in that question now and a lot to be thinking about so I've got one for you to have a go out here it is okay so we're going to go over this question but have a go okay see how many vectors you can label it's asking you to prove that X Y Zed is a straight line so draw that straight line in okay so if you sketch this diagram draw your straight line from XYZ and then think about those three vectors there so you've got X 2 Z being the full line can you find that one out you've got X 2 y which is part of the line quite similar to the last one we just did and you've also got from Y to Z only have to find two of those but you can see how many of those you can actually find okay it's a little bit of hint on that on there but have a go the one thing that you do need to do here is also label the vectors on it tells you what they are here so if we start labeling those up before you have a go o to a it says is a it says o to X down here moving from Oh down to X that's not very good there o down to X is to be and it says from X to B so from X to be this but here is one B so that's everything to label on there everything to be thinking about so pause the video there and have a go okay let's all look at some of the answers then so from X desert hopefully that you've seen that one's not too bad okay to get from X to Z but let's have a look it says a is the midpoint of a Zed so if a is the midpoint then this must also be a and it says y is the midpoint of a B so a B so hopefully labelled on of these are halves here okay there we go midpoint half and half so to get from X does that it's not too bad we just have to go and I'll do this in a different color so look we go up two and then across there from o along to Zed can't draw I think so I'm at a space but we can actually draw this vector so if we draw this in let's write this down from X to Z that is minus to be moving backwards through that to be plus those two A's moving along that way okay so again you could write this as 2 a minus 2 B but ultimately we just want to factorize that now as well you might have left it to the end of factorize but you would have to lots of minus B plus a okay so it's we label that up here we've got two brackets minus B plus a okay obviously you could find some of the other vectors here you can either move from X to Y or from Y to Z now we could have a look at finding all of them but let's just have a look now firstly things first I'm gonna I'm gonna move this way up the line so I need to figure out the vector B to a and then I can do half of that so the next one I'm gonna write down is beat away there we go so to get from B to a we would have to go minus 3 B's up this way plus an A so minus 3 B's plus a minus 3b plus a okay and we want to go half of that so beta y so beta Y will be half of that let's write that one down because we just want to move up to the midpoint so half of that so half of minus 3 B plus a and if we expand that out what do we get minus 3 and 1/2 B plus 1/2 hey there we go and there's our vector v2 y now we can move from B to y we can get from X to Y so let's have a look at that one so from X to Y we can go down that B so B and then add in this vector here so minus 3 and 1/2 B plus the half a and if we simplify that all down one B take away 3 and 1/2 B as we saw on the last one is minus 1/2 B so minus 1/2 B plus that half a and again we can factorize that so if we factorize 2 half out we get 1/2 minus B plus hey there we go and there's our vector so it's 1/2 minus B plus a they go hopefully you see the similarity between this one and the last one but a blank question looks a little bit different asking you to do the same thing but word differently with different letters but actually let you get the same answer for this question so hopefully that just shows you and have we been out of practice and have a go now what you might have seen is that and to be fair before we move on let's just highlight this look we have the minus B plus a in both of them so you would make a little statement there it shares that common multiple that a common vector of minus B plus a but if you did find white a Zed which you could have done we would do a very very similar thing here look we already have the vector for moving halfway up B to a from beta Y and that's the same as moving from white away so that beta Y and let's just have a look at least in a slightly different color because I'm just going to go beyond this a little sec so this vector here that we worked out the minus 3 B plus half a if we wanted to do Y to Z which was our other vector that you could have found we would have had this vector from above so minus three and a half B plus a half a that get us from Y up to a and then we'd have an additional a here to get from a to Zed so an add another a in and if we combine that all together we get minus three and a half B's and then a half a plus one by one a is three and a half a okay and if you factorize that one which we can bring up here you can factorize three and a half out so you get three and a half and then you would get minus B plus hey and again you'd have that matching bracket so you can do all three all three of them work but hopefully that just shows you there how you can actually find the other one as well now one more question for you that's gonna be a little bit different now obviously that one was very very similar to the one we just did but let's have a look at this last one okay so a different question here you could already hopefully see in that ratio it says D is the point on BC so it's at B to D D to C is in the ratio three two one so similar to that first one there you've got you're gonna have to turn into fractions you've got partway up a line rather than halfway and then it gives you some of the vectors a 2b is 3a that's on the diagram eight to C is to B again is on the diagram and it says B to e that big line at the top is three lots of A to C and then it says prove that adde is a straight line okay so if you want to just have a go just there pause the video we're gonna put a few hints on here before asking you to pause it again but if you want no help at all you just want to have a go and see what you get pause the video there okay though if you want some hints here we go so D is the point on B to C so it's that B to D D to C is in the ratio 3 to 1 so that's in quarters so that there is 3/4 and that there is 1/4 of the line it also wants us to find a de is a straight line so I've put my little straight line in and then we think about those three vectors that I want to find so you've got the full length of the line which is A to E you've got part of the line a little bit down there which is a 2 D and you've got the length of the line going from D to e so you need to find 2 of those vectors okay up to you which do you find but you need to find some of those anything else that we could have here is any from the question we've got that last little bit of information that says BT is three lots of a to see why later C is to be down on the bottom of the triangle and three lots of that would be 6b so that bit there is 6b okay so there's two things I would suggest finding first one of them is the vector from A to E and the other one is the vector moving along that line there B to C and it doesn't matter which way you go I would maybe do see today I think I would find so I would probably go from C to B find that vector and do a quarter of that to find out how do you how you get from C to D and then I would compare those ones moving in that direction they go if they see hints if you want to pause the video there we'll go over the full on Serena SEC okay so the full answer for this D is on the point B C so B to D DC's three two one we've labeled that up so we have three quarters in one quarter and we've labeled everything else on the diagram and we gotta prove that ad is e is a straight line so the full vector from A to E is quite a nice one to get from a to a you move up three A's and then along six B's so that's quite nice let's just write that down so it's 3a plus 6b and in fact let's write that down somewhere else because we want to factorize it over there so if we write that down over here a to e is 3a plus 6b and if we factorize that we can factorize three out and it would leave it as a plus 2b okay so we have three lots of a plus 2b and there's that one right so we're looking for this a plus 2b in another bracket now I'll party find both of these but let's go for a e to D to start with so I need to find C to B now to get from C to B I just go - to B so C to a minus 2b and then add 3 a's to that moving from A to B so I get from C to B and we want to do 1/4 of that to get from C to D so from C to D and let's do a quarter of that so 1/4 of - TV plus 3a every time same both by 1/4 we get minus 2 quarters B and you know I'm not gonna write that as a half I'm going to leave it as a quarter amount of quarters in my hunting Minna sec let's have a look I've got minus two quarters minus two quarters B and then plus three quarters a and that's my vector there from C to D obviously we want a to D so in order to get from A to D let's have a look to get from A to C we do A to B so we've got to be we're gonna add to that list of the one here I'm at first piece is negative so I'm not gonna bother putting a plus so we've got to be take away two quarters B and add to that three quarters a now this is sometimes where it throws people but to be I know mama D to be take away two quarters B we could work the boxes to take away a half which is one and a half but I want to leave it in terms of quarters cuz I'm gonna have to factorize it in a sec that's going to make it a bit easier to do that so in terms of quarters to in terms of quarters would be eight quarters 8 divided by 4 is 2 so it'd be eight quarters so instead of thinking about to be I could write it as eight quarters be and eight quarters be take away two quarters be leaves you with six quarters B so we have six quarters B plus three quarters a again you don't have to do this but I just think it makes it a lot easier here to actually think about how this factorizes because now we've got the both of denominator as for as for you can see that they both divide by three quarters there so three quarters fits into both of them and if we factorize that let's see what we get we get three quarters and three quarters fits into six quarters twice so we'd get 2b which is what we want it matches up there and three quarters into three quarters is 1 so plus a and again my vector has actually come out in the opposite direction now in opposite direction the opposite way around because up here it was a plus 2b and here it's 2 B plus a means the same thing so we can just rewrite that as three quarters a plus 2b there we go if we label that up here we have three quarters really about 3/4 then let's try and redraw that 3/4 any plus TV and there we go and it does say prove so we would have again one I just write a little statement here just centered they are both multiples of a plus 2b both share the same vector they're moving in the same direction and if they're attached at D and they have to be on the same straight line because they are moving in the same direction have that same attaching point there but just obviously as a side note we could have found D to e so if we could do that one as well it's a little bit more complicated on this one because it's not half in half so for C to D is 1/4 of that I'd have to find 3/4 of the line now which wouldn't actually make too much hassle here but if you did do it your way around it would just be at this point down here when I did C to D so ever look that one there I would have done 3/4 of that vector and if we work that out and so I look 3/4 and see if we can work this out to the side run out of space here let's get rid of this vector a 2 over here and see if we can do it here know again so if I did do D to e DD I would have to do 3/4 of C to B so I would do 3/4 of minus 2 B plus 3 a and then once I've done that I'd add in your 6b to get from B to e so once you've expanded that you would then add 6 B's in and if we work this out myself like what do we get we get minus six quarters B plus nine quarters a3 times 3/4 plus the 6b and again how many quarters is 6 that is 4 times 6 which is 24 so be twenty four quarters B and if we join that all together 24 take away the 6b at the start they just with 18 B again not got no space here so we're right over here we've got 18 quarters B and nine quarters a there we go and again we can factorize that by nine quarters this time so if we take nine quarters out just about fit this in we get nine quarters to be plus a or a plus to be there we go just about fit that in sorry it's quite small on screen there but that is just an additional step and you actually can see get the matching bracket no matter which way you took okay so I always up to do that first part of the line rather than the second part of the line but I've just my personal preference on how I like to do it so I mean there's a lot going on there particularly if you're not that comfortable with vectors okay you might have found that quite daunting doing some vectors there or it might just be a nice reminder for you effect this is something quite comfortable with it is a tough topic though so you know you might want to just go back run through it all again make some notes keep practising those questions and trying to get comfortable with all the different bits of maths going on in this question here but that's vectors I will have a look at some of the harder ones soon and because some of the ones that been coming out quite recently are very nasty there's just about thinking on a question like this how can you find bits of each question that you can get marks on one of those tends to be finding the full length of the line that's always a good one it's never too nasty to find and the other one is finding a vector of a line that has no vectors on it so in this case that was C to be or you could have gone from B to C but either way you want to have a look at what line has no vectors on and see if you can create a vector for that that's the end of the video again if you found that you saw please like please comment please subscribe and I will see you for the next one [Music] you [Applause]