GCSE Maths How to Solve Parallel Vector Questions

hey guys welcome back to another GCC maths video this time we're talking about vectors specifically how we can prove that two vectors are parallel using exam questions but before we get started I just wanted to let you know that the majority of you who watch these videos aren't subscribed so if you've ever found any of these videos useful to you whatsoever if you could please just click the Subscribe button it really helps me out without further Ado let's jump straight into this question so before we get started we have to prove that two vectors are parallel but what does that mean well when you have a vector for example the Vector 1 one or if you wanted that could also be you know a plus b all that means is it's going one unit to the right and one unit up so it's a straight line just going like that between two points if I double it for example and make it 22 which again could be also written as 2 a + 2 B all that would do is it would stretch out thanks Microsoft I appreciate it stretch the line to make it twice as long so essentially when you have vectors if they are scale multiples of each other it means that they are parallel now a lot of people have a bit of trouble understanding what I mean by scale multiples basically if you can factorize out an integer from here so if I look at 2 a + 2 B imagine it's just normal algebra you could factorize that to give you 2 A plus b right if you notice that is this Vector multiplied by two two that means that these two vectors if I want to call this Vector 1 and Vector 2 are parallel to each other and this actually works for fractions as well so again if I have that a plus b and I have something like a half a plus a half B again we can factorize out2 so the aim of this game so to speak is to work out the vector PQ work out the vector Fe and try and factorize it so we can write one vector in terms of another so again for example if it was a plus b and then a half a plus half B my goal is to write it in the form where where I will have a plus b inside a bracket that's the end goal for this so starting off with vectors I always give a few kind of pieces of advice the first thing I'll do is if they haven't written on the diagram what vectors they given you I would recommend doing that first so first things first is FD is a so F to D remember it goes left to write like you're reading a book we're going from F to D so this Vector here is a and then F to e is B so the vector directly along here is B perfect they usually always give you one or two of these normally two to be honest and let's read some other information so we've used both of this information it says p is the midpoint of FD so what that means is each part of this so from F to P is a half a and from P to D is a half a as well because it's halfway between it right so if the full length was let's say 10 cm each part would be 5 cm same thing here and then we also have q is the midpoint of De so again we can say that's a half and a half but I don't know what the vector de is so I can't write anything else and now it's saying use a vector method so that might sound a bit fancy basically just using the algebra I showed you just now prove that PQ and Fe are parallel to each other so what we need to do first and this will always gain you some Mark so this is a four mark question you always get around two maybe three marks if you're lucky to work out PQ and Fe normally it's three marks so P to Q well if I start at Point P I'm trying to get to Q now how can I go there well I can't go straight to the right instead I have to go along these lines okay imagine that you're kind of trying to walk across these lines like roads you have to walk across the road you can't just like create a brand new one in the middle so I have to go from P to D and then from D to Q I would write that out first because I personally think that this it helps me a lot so we have P to D and then we have D to q and then what I'm going to do is I'm going to ask myself well do I know what P to D is well P to D is just a half a we've already got that perfect and do we know D to Q well we don't but I do know that D to Q is half of D to E so I'm going to write half d e remember that the point Q was halfway between D and Q it's the midpoint okay does that help me a little bit because I couldn't work out what DQ is so now I've changed it in terms of De I've got half de can I work out D to e well again if we get rid of those two points if I start from the point D and I end up at the point E I could go straight down here but I actually don't know what there's no Vector that goes from D to e yet right I can't actually just go straight from D to e but I do have a vector that goes from D to F and then F to e so that's the way I'm actually going to go I'm not going to go straight down here because I can't there's no Vector I'm going to go down to the left and then to the right the second piece of information I like to give students with vectors is you have to go along vectors that you know or you can work out I don't know the vector D to e so instead I have to go along the ones they've given me which is D to f f to E so let's do that so we now have a half a because that's from the previous part plus a half and then D to e well I need to go from D to F here's the third thing if you notice my arrow for a is going up and to the right in this case I'm going down to the left so instead of a it's actually going to be negative a remember from science potentially vectors have direction as well as size so it's not just going I'm not just going a I'm going a in a certain direction up and to the right so if I'm going in the opposite direction so again I'm going downwards I need to make it negative instead and then lastly F to e well again I'm going in the same direction as the vector right I'm going from F to e so I'm going this way so it's just going to be plus b then from here we can just expand this in order to work it out I'll do this in a few steps so a half a minus a half a a plus a half B see y half a and then if we expand this we have half * minus a so minus half a half * B is uh half B so in short P to Q is just a half B now for the vector Fe we actually already know it CU it gave it to us in the original uh in the actual question itself Fe is B so all we need do is say that that equals B that right there gives you three marks so you get one mark for writing this General expression over here another Mark for working out what DQ is so in this case we worked out DQ over here the final Mark for simplifying it the fourth Mark is for actually explicitly showing that PQ and Fe are parallel and this is where some students lose some marks well a mark in this case because they don't explicitly show it I'm going to show you how I would do it now factorizing a half B might seem a bit strange but instead I can say that Fe is equal to B so I can just replace B with Fe so that means PQ whoops is equal to a half Fe and again remember if you're factorizing it you'd write it like I did originally so if it was a half B plus half a or whatever You' factorize it then do this and then I'd say therefore PQ and Fe are parallel and that way you can be super super super specific and explicit about why those two vectors are parallel in the most recent exam Series so the 2024 exam series when students got results they showed me some of their papers and a lot of them would lose the final Mark in questions not necessarily Vector questions but in questions because they weren't explicit you have to really show this and I know that when you're revising you might want to skip this because it gets a bit tedious therefore this therefore this therefore this you need to remember when you're in a mock exam or an actual exam you need to be very specific because if if I'm an examiner I'll look at this and go okay oh I see this well their reasoning is good then I'll check they working out if I don't see this I might be a bit more skeptical but hopefully that kind of helps you with dealing with Vector questions especially when they say show that they're parallel remember that there are other versions of this so they might say given that they're parallel find um one of the vectors or find k or something along those lines I will have a video coming on that shortly but again if you found this useful please consider subscribing just to help me out a bit more and I'll see you in the next one