Vectors 2: The Most UNDERRATED Topic in O-Level Maths

[00:00:00]Hello and welcome to yet another exciting episode of maths on unlockwise.

[00:00:06]I am your host Martin Tatina also known as math guardola most students favorite exam coach and today we'll be continuing vectors uh we will be doing vector 2 which is vector geometry right so it's more of a continuation of vector one which is normally examined in paper one u so if you haven't seen that video I did uh be sure to go on the playlist of all level mathematics and find the video whereby I'll be teaching vector one that will help you to understand more uh of what we'll be doing today right and I also expect you to know how to solve simultaneous equations we want that knowledge as we will be proceeding right. So today we will start by finding a vector like we will be expressing a vector how a vector is expressed uh geometrically on uh on a diagram. So you should be able to understand diagrams and vectors right? Then we are also going to look at ratio and then we will also look at constants. Uh then we will also find the constants h and k. Uh that's where the simultaneous equation knowledge would be required. And as I was preparing for this lesson uh some students requested that I took special emphasis on that. So I will do as you requested. Your will is my desire as well. So I will be delivering your will. Right? I am a listening teacher and a talkative as well. Okay. So after that we will also look at ratios of area of size. Uh somebody also requesting that. So I will make sure that I specifically put a lot of emphasis and try to make it as simple as it should be. Okay. So uh without further ado I want us to dive into what is known as expressing a vector. But if we'll be doing expressing a vector in our previous lesson whereby we were doing vector one I introduced what is known as displacement vectors.

[00:02:52]Right? And as we were doing displacement vectors I mentioned that the direction in which our vector is given is very very important. Why? Because by definition a vector is a quantity with both magnitude and direction which mean that the direction part does matter is even part of the definition right so I will be using uh this diagram here to explain a few things that I want you to understand before we dive into the vector geometric 12 right so we have what is known as the triangular law of addition triangular law of addition right so what I'm going to do here I'm going to explain what this means right we have O A which is this part O A and the way you are given this is O and there is this sort of arrow which is showing that we are moving from O to N. That's the direction part. You see that's the direction part. And I also want you to notice this vectors can also be represented by an A like this or you can have it uh being bold. Right?

[00:04:35]Just trying to make it bold.

[00:04:37]That's what you see here and you understand. So it can be written as this a and then we put uh a dash on top or in some text book you see it like this and maybe some zigzag down here or a straight line as well a dash no problem it's one of same thing so okay let's continue right so from O to A right which mean that we're moving from O to A then plus A C which means that we are moving as well from A to C and we are now told that this will give us the vector of C right it will give us vector O C and it's sort of a triangle, right? If you are to check this one, this one to give us [clears throat] this one, right? So let me maybe draw it in miniature.

[00:06:04]So A O A C. So from O to A and then from A to C will give us vector O C right. So this is what we refer to as the triangular law of uh addition when we are doing factors.

[00:06:30]And I also want you to notice this since this is a parallelogram the opposite sides are parallel and equal. So I will specify that when I am explaining parallelogram law of addition right okay but I just wanted you to uh no advice right so let me erase this and then um explain the paraloggram Know what?

[00:07:25]Basically you should know the truth and they are not very easy. They are not very hard to understand though. Right. So we move on to the parallelogram law of addition.

[00:07:42]Right? So what happens here is parallel sides or opposite sides are parallel and equal which means that OB is equal to AC and BC is equal to O A as [clears throat] well which means that they are both parallel and equal. You get it? That's one of the properties of a parallelogram. So we will utilize that knowledge when we are dealing with parallelograms.

[00:08:17]Right? So what is going to happen here is uh if we want to find OC right OC is going to be equal to O A [clears throat] from O to A and then we add from A to C.

[00:08:39]Right? This is going to give us O C or we can write it as O B plus B C which is also equals to OC. Right?

[00:08:58]So why am I telling you all this is because in some cases you will just be given O A and O B and be expected to give uh in terms of A and B vector or C right and well lately examiners have been obsessed with parallelograms.

[00:09:33]You can check June 2024, November 2024, parallelogram parallelogram. Got it? So uh I managed to explain triangular law of addition because most of our vector questions are in triangular form maybe uh trapezium something of your sort but lately we've been seeing parallelogram parallelogram so I also managed to include paralle Right. So I also have a question here from June 2010 paper two number seven.

[00:10:21]So that I will use it as reference explain a thing or two as we will be going forward. Right? So having mentioned this I will come to the thing I mentioned as expressing a [clears throat] vector. That will be the first concept I will explain. I say that I will be explaining about five concepts. Right? So this is one of the things that will help you understand uh expressing effect. Right? So under expressing a vector what we are going to be looking at really is the knowledge on displacement vectors.

[00:11:06]such that we we make sure that we understand that if we are going the same direction as the arrow then the vector is positive. If we are going against an arrow the vector is negative right? So this is my first concept by expressive vector.

[00:11:41]So right now let's say that you have a diagram something like this. Uh we have four here and a we have B here and you are asked to find A B get it? Now we're just going to we know that movement from O to A okay let's call this A like this and movement from O to B let's call this B like [clears throat] this right and we want to find the vector A B in terms of A and B like this so if We are moving in the same direction as the vector as the given vector it's going to be positive.

[00:12:50]Same direction as the given vector is going to be positive. But if we are moving against the arrow of the given vector then we multiply the vector by1 or by a negative or the vector result is negative depending with your own understanding.

[00:13:14]That's the thing. So allow me to do this.

[00:13:20]We we we want to have vector AD, right? So we don't have a direct link of A to B right. So we are going to use the given vectors. We move from A to O then from O.

[00:13:47]So in other words a vector a b is equals to n o plus or b like this right? So what is going to happen here now is that we were given O a not a O. So we are going against the given vector and once we move against the given vector the vector becomes negative. So in other words I'm having O A plus from O to V we are moving in the same direction as the given vector which means that this is going to be positive. So we are going to have positive or plus O b right let's hope this is clear right so what I want you to see is we are going to have a plus b which is positive by the way. Okay.

[00:15:05]So, some even prefer to write this as uh B minus A starting with the positive one. Well, uh it's up to you.

[00:15:22]Personally, I prefer to leave it like this because it shows me the movement I actually used, right? But It's still okay. They will mark this right. Uh it's not an offense. It's mathematically correct. Uh there is a language like mathematics whereby they say is commutative, right? Uh 2 + 3 is the same as 3 + 2. That won't outer your answer.

[00:16:00]So no problem. If you want to write it like this, uh it's even allowed right.

[00:16:06]So from here now I will give you some questions which I will be expecting some answers in the comment section. Uh I appreciate a lot of students who will be in the comment section asking questions responding to questions I would have posed and uh it makes us interact more than just on this video because like right now I'm the one who is speaking and you're doing much of the listening but for me to understand for me to get that you are understanding what I'm I also want you to respond in the comment section so that we can interact and uh maybe clarify some other misconceptions that may arise during this lesson. So I want to give you a small exercise.

[00:17:12]So my first equation will be this one and this one will be a [clears throat] right.

[00:17:36]So I want you to find vector B A and then I also want you to find uh vector A.

[00:18:00]This is my first question.

[00:18:03]Then my second question will be something like this.

[00:18:16]Then this is P Q R and this will be 4 A.

[00:18:32]This will be 5B.

[00:18:37]Right. Put some diagonals and then have this as 3 A.

[00:18:46]Then I'll ask you to find 1 o 2 r t and uh I will also ask you to find pq right so you can copy And I will give you a third one which is on a parallelogram so that you be able to blend what I've taught you prior and uh what I was teaching now may you use the parallelogram rule of parallelogram law of addition uh as taught right okay so you You can even pause the video, copy this and then uh continue. Right, [snorts] having said this, [sighs] let me give you your number three.

[00:20:05]Question number three, something like this.

[00:20:19]We have A B C and maybe O here.

[00:20:30]Okay.

[00:20:32]So, O A is 3 [snorts] P - Q and then O C is P + 5 Q and then let me give you the questions. So, I'll just write a brief description here in the diagram.

[00:21:07]O A B C is a parallelogram in which uh O A or vector O A is equals to 3 P - Q and O C is equal to P + 5 Q uh Express as simple as possible one uh vector BC.

[00:22:20]Then uh I also want you to express uh vector AC. Now what I want you to do here is to express BC and AC in terms of uh P and Q. Right?

[00:22:40]given that this diagram is a parallelogram that's one and that O A is 3 P minus Q and O C is P + 5 Q right I'll be expecting your answers in the comment section below and I will be marking and giving feedback as well right I want you and I to interact that is the major purpose of giving you this exercise right so done with this uh let me step aside and then maybe you can copy this for a moment but having done this I also want to move on to the next aspect which is ratio right uh our next aspect which is ratio right so You're going to aspect number two ratio.

[00:24:00]So in our questions you'll be given sides vectors with their ratios. Right? If you remember quite well in our vector one session in our vector one lesson I mentioned that parallel vectors uh it means that one is a scalar multiple of the other which means that there will be a ratio there right you can have the sides being scaled like one is a scala multiple of the other right uh in this case I will even explain using midpoint but if you were to check on this question as well I want to include to our lesson right in the diagram PQ is paral to I right we are using parallel vectors as I was saying uh which means that this one and this one are parallel and one is a scalar multiple of the other. All right.

[00:25:16]Fine. Then you are even told that P M is equals to 13 of P. Right?

[00:25:27]We are talking ratios.

[00:25:33]Anyway, I I will even try to simplify uh some of the questions here. I will answer some of the questions here and help you apply the knowledge I am giving to you. Right? And maybe I will start when I reach uh con state then uh I will also land in the question when we reach a point of finding constants because I know that most of you that's where they begin to get better but I will try to break it um into small chunks you can right. Okay. So anyway [snorts] let's say to our initial back to our initial diagram which was like this when I had four here a here B here like this.

[00:26:41]Maybe I can put a line here and then call this right. And now uh if you are given uh that O A is equals to A. So I'll put N here and uh OB is equals to B.

[00:27:10]Right.

[00:27:12]And that m is the midpoint of a right.

[00:27:32]I want you to express a m in terms of a and b. I love this.

[00:27:58]I will explain why.

[00:28:02]But we have our A here and our B here.

[00:28:06]So I want you to notice something, right?

[00:28:09]Uh we are given that O A is A and O B is B and that M is the midpoint of A right vector M. Let me do this so that we appreciate how vectors are given. Right?

[00:28:36]Okay.

[00:28:38]Uh what I want you to understand here is we can find vector A that's what we need to do first right and I haven't taught midpoint of a vector in the first lesson when I was saying yet is yes introducing scalar has the given vector right so we will find the vector AB and then introduce color. Let me use a different color so that you appear different and you understand what I'm doing, right?

[00:29:19]Uh so I want to look for vector A vector A. It mean that I'm going to move from A to B. But there's no direct vector here.

[00:29:29]So we are going to move uh AB is going to be A O plus O B. I think I have managed to explain this. And then what is A O? A O is minus A. And what is OB? OB is positive B. So this will be my value of uh a but now we want the midpoint which means that uh midpoint of this one which is actually a m right a m is equals to half a right and uh what is ab is - A + B, right? And uh you can leave it like this or you can expand it. Uh this will give you rather uh for those minus half a plus half b. And then for those who love to start with the positive one, you even write it as uh half b minus half a. Yeah, I think I get the point now. This looks less creepy than having a negative uh in the beginning and uh it's okay. All right.

[00:31:22]So, let's hope that this is making sense to you and uh Okay.

[00:31:30]So I will have to maybe attempt the question I have uh like prepared for you here so that we'll understand each other especially here where we are asked to look for PM uh yet PM is one uh PR are. So, let me answer this question for you and uh let's hope you'll be able to follow along. Right. So, we have this nice diagram here and then there is a brief description and I urge you to read this description very well if possible twice or thrice. Why?

[00:32:29]Because there are some contents which you cannot see on the tag that are described in the description. Let me repeat this because this is very important. I don't want you to miss this.

[00:32:46]You have to read the description at least twice because there are some contents which are in the description but cannot be seen on the diagram. In as much as the diagram is a clear representation of the description, there are some things which are on a clear which are not on a clear representation of the description that you get from the description. So you need to read the description.

[00:33:26]Just guess. Okay. So in the diagram P and Q is parallel to PQ is paral to O R.

[00:33:38]Okay. So PQ is par to R.

[00:33:44]Unless stated don't assume.

[00:33:49]Unless stated don't assume. Okay. Then we are also told now that PM is equals to 13 of P R. This is where I want you to understand.

[00:34:07]Let me get a little cheeky with math, right? I want you to get what is missing on the diagram, but I'm getting it from the description. If PM is 1/3 of PR, it means that this one is let me just 1 / 3 and this remaining part will be 2 over 3. such that the whole line will be 3 over 3 which is one is one of Peter which means that from here to here is one of the from which mean that if this is one then The remaining part is two sets, right?

[00:35:18]We just remove one from a wall, right? And we remain with uh that's where the matrix starts to come into play. That's where a lot of people will get confused, get lost, but it's as simple as this. Right? So having mentioned this, I'll also try to break it down as we go so that you understand.

[00:35:47]All right. So anyway, right um OP which is this one is 2 A and O R this vector is 3B. Right? So express in terms of a f or b one vector p r i. Okay. So what I want you to see is we are asked to express in terms of a or b the vector p r. So P R is equals to P O plus O R and then P O is identity the given vector which is O P. So I won't spend much time explaining that. I'll be just writing and if you are still confused you can ask me in the comment section.

[00:36:53]Right. So we have this one is our P O and we want O is this one.

[00:37:06]So [snorts] O R is in the same direction is the given vector. So we have this as 3B.

[00:37:18]So you can write it this way or like I mentioned that some will prefer having it as uh 3 b minus 2 a no problem as long as uh it's one and the same thing just like as it is okay right we move on to the next part so maybe I will show them here so that I will see them and uh maybe use them future.

[00:37:54]Um so this is - 2 a plus okay so I can come back to this and explain as we go. Okay.

[00:38:22][snorts and clears throat] Uh so that's it. Okay, let's move forward.

[00:38:32]We are also asked to express in terms of A and B uh vector PM. Okay. Right. So what I want you to see now is vector PM.

[00:38:50]Vector P M is 1/3 of P R we've calculated P R and we are now supposed to calculate 1/3 of PF. So PM is equals to 1/3 of P right no problem. [clears throat] Uh so this is going to be 1/3 of whatever value you have which is - 2 a + 3 b right and uh I'm sorry to those who don't like fractions I used to hear them too but sometimes you cannot run away from them. So 1 three you get 1. 1 * b you get b talking.

[00:39:54]Okay. So yes I multiplied once by -2. So it's something like 1 / 3 * -2 and any number can be over one in numerator time numerator you will get uh -2 and then denominator* denominator you'll get three. So this is how I got my -2 uh over 3. And then uh for those who are wondering but while when doing this I expect you to be in those top classes whereby this won't be a problem. All right. Uh so this is over one. If you want you can even cross this one and this one and get 1 * 1 which is one and get 1 * 1 which is 1 and 1 / 1 = to 1. A lot of ones here, right? And uh maybe if you want as well, you can have it like uh maybe [snorts] you can have it like this. 1 / 3 * 3 over 1. Then 3 * 1 you get 3. uh 3 * 1 you get three and this will give you one. Yeah. So this one in the same thing but anyway if you want a less construction you can also request in the comment section below. Right. So uh we have managed to deal with um PM which is this one and allow me to maybe just keep it somewhere. Why? Because what happened with vector 2? One of the things I love about questions uh asked in factor 2, they keep on accumulating from what we have already got like how we used PR to get PM and how we will use PM to get maybe OM right as I want you to see right what is the value of uh PM is uh something like - 2 over 3 a + b. Okay, that is it. So to erase this so that we can do what we are doing. Right. So we are now at a point where we are supposed to express O in terms of A and O B. Right? So now I want you to see this uh OM we move from O and we want to reach here. So how are we going to do this?

[00:43:25]Okay, we are going to move from O to P and then from P.

[00:43:38]Yeah, I hope you're seeing this.

[00:43:48]This is our movement.

[00:43:53]It's okay.

[00:43:55]Okay. So, OM is simply OP plus PM, right? OP we are given already. This is 2 A and then PM we have calculated it here.

[00:44:19]uh and pm is equals to -2 / 3 a + b.

[00:44:29]So if I remove brackets by multiplying by the fraction by the sign bracket I will multiply the sign with the fraction first plus and minus will give you minus and uh this will remain as 2 over 3 a plus and plus to give you plus. to have SSB no problem right and then uh let's just simplify the a bit so that it won't look creepy right we have 2 minus 2 over 3 so uh this should give me 4 over 3 a plus B.

[00:45:27]Oh, let me maybe break it down a little.

[00:45:32]Okay.

[00:45:33]Uh, if we have 2 - 2 over 3, maybe you want to have a common denominator.

[00:45:45]And then 1 into 3 is 3. 3 * 2 you get six. And this will be 2. And if you separate you have 4 over 3. So this is it right.

[00:46:01]So fractions will still help.

[00:46:09]Okay. Right. So this is my answer for OM. Right. and I will keep it because as we have seen the answers may still be required.

[00:46:24]Okay, especially that OM. Well, okay, let me keep quiet. I I will remind you as we go. Right. So, we have our OM as uh 4 over 3 a + b.

[00:46:46]And I want you to keep this answers because they will be required as we go.

[00:46:53]Okay.

[00:46:54]So, uh, we've reached another point in this lesson.

[00:47:03]I think this is the easiest, one of my favorite. I love easy things, right?

[00:47:14]Hi.

[00:47:15]>> Look what Okay. Right now on this aspect we are just going to introduce the constant nothing else just introducing the question right so uh uh let's see uh a given that pq is equals to h or r Right?

[00:48:05]We have PQ which is equals to H O R.

[00:48:13]Right? So what I want you to understand here is that PQ is equals to H O R. So PQ is equals to H. We have our O here right as uh 3 B right and then you can even remove the bracket and have it as 3 H B.

[00:48:44][clears throat] Uh I know a lot of people are going to question me here. uh want to understand why I actually had to write it as 3 HB instead of 3 BH because normally we write the things in alphabetical order right so in my response in my defense as I will be doing vector geometry with you uh we will be expressing vector vectors uh in terms of a and or b as we are told. So uh we will consider this part to be the coefficient of b [clears throat] and uh sum to be the coefficient of a as you will see. So what I'm going to do is I'm going to answer the B part one part uh the A part and the B part and then we will do uh finding the constant right.

[00:50:05]Okay. Uh so basically from what we are given here our question states that given that PQ is equals to H O R. Write down in terms of H A and O B an expression for oxygen. Before I go further here, I want you to understand this.

[00:50:34]When they introduce a constant and tell you that it is given this, right? That statement you have been given, you have to use it to find whatever they are going to ask.

[00:50:49]Let me repeat this.

[00:50:52]This PQ is going to be used to find maybe uh this PQ which is equals to H O R will be used to find PQ in the first part and OQ in the next part. You will see this as we go right. So maybe uh we have used this expression PQ and we now know that our PQ is equals to 3 HB and I have explained the issue of me having it as 3 HP. Right? Okay. So fine I will erase it here.

[00:51:41]But I want you to keep it in mind, right? Um, okay. Maybe let me write it here like this. uh P Q which is equals to O and uh this is the value right. You are now asked to maybe express uh PQ in terms of H and A and R. I think this is the answer for this one. So, okay. Uh let me let me say is I is not what I think that is the answer. Okay.

[00:52:33]So, the answer here is supposed to be 3 H. [clears throat] Okay. fine.

[00:52:42]Then um from there we need to find or Q right. So what I was explaining earlier on is that you need the PQ for example.

[00:53:09]Okay.

[00:53:10]How? So, OQ is equals to O plus PQ and you want to express this in terms of A, H, A and B. So, OP no problem. You have it as 2 A. Then PQ we have just three B right. Okay. So this is what we have now. Right. [clears throat] Uh so allow me to keep OQ as uh 2 A + 3 H B right that's our value of O right So this is the part of introducing constants. Now we want to move on to one of the requested part by many students. I mean many students uh for you to understand where I am going to be going explaining a lot. I think it will make sense for me to start by answering a question that was in June 2017 paper one number 26. It reads like this. Uh okay, let me first write this down like uh we are doing what is known as finding constant.

[00:55:37]My favorite part. Okay. So it is given that O is the origin.

[00:56:02]4x is equ= to 2 a + 3 b and uh all y is = 3 a - 4 b then you are asked a to find x five. Well, this we have passed it and then you are also asked uh to calculate not to calculate as such. It is also given that it is also given that Ox is equal to let me write it in a straight line so that you see what I'm trying to mention.

[00:57:22]Ox is equals to 1 - h a + k b. uh then you are asked to find the value of uh H and K. All right. So what I want you to do this is maybe copy the question or maybe if you have uh the question paper uh this is now June 2070 paper one or number 26 six right uh so one of the things I want you to understand is we have already discussed about this part whereby you are asked to find uh xy right maybe what I'll do is I'll just uh answer it quickly just run through but my main focus now is finding the constants right and I want to simplify it so that you understand this better okay so let me remove this so let you see what I'm doing okay right so the first thing I'm going to do is I'm going to draw a sketch let's say this is my core And uh this is my y this is my x and uh okay fine. So according to what we were told x = 2 a + 3 b or x right is = to 2 a + 3 b and y = 3 a - 4 a right and now we want to find xy so we are going to move against this arrow But please bear in mind that this is considered as one term. So we'll put it in brackets. Minus 2 a + 3 b.

[01:00:09]And then from y is going to be 3 a - 4 B right.

[01:00:30]So is something like this.

[01:00:35]Uh you remove the brackets by multiplying everything by -1 is okay.

[01:00:45]So this becomes -2 a - 3 b + 3 a - 4 b.

[01:00:55]Okay.

[01:01:06]Okay. So if we group like terms we are going to have uh - 2 a + 3 a - 3 b - 4 b and this will give you uh something like a - 7 b right so this is the first part and uh since we have discussed this way back. I'm not going to maybe dwell on it much, right?

[01:01:48]But my main one, my main uh priority here is for me to show you what I'm about to tell you. So, please pay attention.

[01:02:06]Please listen attentively because this is where it's about to go down. Right?

[01:02:15]So we have Ox as um 2 A plus 3B and we are also given O X to be 1 - H A plus K B.

[01:02:42]Right? I invite you to >> listen to what I'm about to say and listen carefully, listen attentively, right? I think I even repeated this so that you open your ears uh you know uh to your best of your ability right so we are at the point where we are supposed to find the value of h and k this is where I was saying that simultaneous equations will come into play so I want you to look at this uh and understand it right.

[01:03:26]We are going to use a method of comparing the coefficients. Right? We are going to compare the coefficients.

[01:03:36]What do I mean when I say right? This is O X. Let me reduce the font a bit so that I can write on it. This is Ox right?

[01:03:48]And this is also Ox right.

[01:03:57]And all things being fair and square, we are told that Ox is equals to this.

[01:04:05]And we also told that Ox= to this which now means that 2 a + 3 b is equals to 1 - h a + k b.

[01:04:23]This is where a lot of teachers struggle to explain and a lot of learners begin to hate vector geometry.

[01:04:36]So I want you to listen to this and I will try to explain maybe twice thrice so that you understand right. I want you to see this.

[01:04:50]Since this is equals to Ox 2 A + 3 B is equals to O X and 1 - H in brackets A + K B is also equals to O X. It means that these two are equal. These two are equal. That is reason why I have to equate them like one on my right and the other one on my left. Right?

[01:05:16]But I will even explain using one on top of the other like this so that you will understand and you get the picture clearly as I want you to do. Right? So in terms of a two is our coefficient on the left hand side right.

[01:05:42]If we have two here and have a here and have this one here 1 - h and have a a is common which now means that 2 is = 1 - h let me try to do it Yeah, [clears throat] we have our A here. We have our A here.

[01:06:15]Okay.

[01:06:17]And we have this two here and this term here.

[01:06:23]So 2 is = 1 - H.

[01:06:33]That's the tip.

[01:06:35]Okay, then let's move to the other part.

[01:06:51]We have B here and we also have B here.

[01:06:56]And we have coefficient of B being three and coefficient of B being K.

[01:07:04]Which now means that three is equal to K.

[01:07:14]That's how simple it is. But maybe uh sometimes our teachers, our educators, our facilitators, our tutors, our exam coach may fail to explain it in a way which uh you can comprehend right and let's hope I'm not trading that right.

[01:07:41]I'm trying my best to make it easy, to make it simple, right?

[01:07:46]uh like I'm saying this we have B and we have B right and this two things which normally said three should be equal to K that's it right so if three is equal to K then I also want to find the value of is right but already in this example uh k is equals to 3. So it's okay. Uh I think I've highlighted it here. Uh 3 is equals to 10. So what I want us to do now is to find the value of h.

[01:08:40]Right?

[01:08:43]So if 2 = 1 - h and what is the value of h?

[01:08:50]You might even uh put the h on the side and the two on that side.

[01:08:56]>> Yeah.

[01:08:58]Okay.

[01:09:04]Forgive me if I have done this any way you might not have appreciated. Let me do this maybe.

[01:09:13]uh cuz I don't want you to get lost. This one comes on the side so it becomes minus so it becomes - h and then this becomes 1 is equ= to minus h you divide the side by minus one multiply by minus one whatever and then this one and this one will cancel and then this will give you -1.

[01:09:38]So h is equ= to minus1 which is here right. So uh that is the thing. So uh uh my answers are h therefore h is = -1 and k is = 3. Okay. So let me do this.

[01:10:05]Let's hope you've seen what I was doing and uh now we want to apply it to our question.

[01:10:14]Yeah. Right.

[01:10:18]Um given also that OQ is equals to K O M.

[01:10:28]write down another expression of OQ in terms of A, B and K. Okay. So if you still remember okay uh we have O being equals to K O M right. So we want to write another expression of q uh in terms of k a and b. And if you still remember I even told you that uh please keep the value of this am right.

[01:11:11]Okay.

[01:11:13]Now so we want to write in terms of a m. Okay. So already we have the cake and we want the value of OF of which is 4 over 3 A plus B. Okay. So remove the brackets you have this as 4 over 3 K A plus K B. All right. And I think I explained why I'm not using alphabetical order in this case. But under normal circumstances, please write this letters in alphabetical order and maybe allow me to say this is not a normal circumstance.

[01:12:10]Okay.

[01:12:12]So this is now my other expression of uh OQ, right? Uh allow me to write it here as well as uh 4 over 3.

[01:12:39]Okay.

[01:12:41]A + K B.

[01:12:49]All right. So [clears throat and snorts] this is where it gets interesting.

[01:12:57]We now have two expressions of all right. one we got uh maybe here uh so let me write it down or Q maybe let me increase the form a bit Q is equals to 2 A + 3 is B.

[01:13:40]And we also have an expression of OQ which is the one we just got. 4 over 3 K A uh plus K B.

[01:14:02]Right. So we are now going to do what I was saying uh is called comparing the coefficients.

[01:14:14]So this is what we are going to do now.

[01:14:18]We have a here and a here.

[01:14:22]Everything before a is equal. That is the reason why I have to put my k before A. So that when I start to say that this one which is before A is the coefficient of A and whatever is before A is the coefficient of A as well. You understand that 2 is = 4 over 3 K.

[01:14:53]A is common.

[01:14:55]So it's okay. So this will be your first equation.

[01:15:01]And then we also have here = um 3 H is = to K, right? And this is now your equation two, [clears throat] right? So having done this, you now have two equations. You can calculate the value of K and the value of H simultaneous equations. Right? So allow me to write this down and then solve the simultaneous equation for you and then uh we will redis and uh maybe I can emphasize how you will get this coefficients because this is where a lot of student feelings get lost Right? I think it's easy. I I'm making it look easy, appear easy, right?

[01:16:10]Because it's not hard. It's not even easy.

[01:16:14]2 is = to 4 / 3 k and uh 3 h is = k.

[01:16:26]Okay.

[01:16:30]All right.

[01:16:40]So now we want to solve simultaneous equations. I think a lot of you will be back in this game. Okay. So for let's say we have 2 being = to 4 / 3 k and we have this as 1 and we have 3 h being = to k and we have this as two.

[01:17:10]uh now we can solve simultaneous equations in a lot of uh ways but uh I think for this session I'll just take this one and use substitution method. So we are going to have this as 2 is = to 4 over 3. Then the value of k is 3 h. So we have this as 3 h. Right? So three and three will cancel each other. So we have this as uh 4 is equals to 2. And then I will divide both sides by four.

[01:17:55]So it's by four.

[01:18:02]So my h is equ= to 2 / 4 which can be reduced to 1 / 2. So h is equals to half. Right? So now having found the value of h allow me to keep it here. h is equals to half.

[01:18:26]Right? Uh I can also uh calculate the value of K. Now 3 H is equals to K. And what is the value of H? The value of h is of.

[01:18:59]So what is the value of k? You multiply this and you get 3 / 2 which is equals to k. So k is = to 3 /2 which is something like 1 and a2 right. So my k is equals to 1 and a half. Right? So this is uh how we calculate the constant. Right now uh I'm left with explaining how we find the ratio of areas.

[01:19:46]Uh so since this has been a long video uh too much for your concentration span but I think I've managed to cover the most important concepts. uh we are left with a ratio of areas and whatever which is uh out of the 12 marks it's just a single mark right so what I will do is in my next video I will touch on ratios area of ratios and then do past exam questions with you so that uh it sticks better and we understand vector geometry to the fullest. Right?

[01:20:37]So uh let's hope that you managed to understand what we were doing and if you've been watching until uh this present moment I would love to thank you for your eager your zeal to learn vector geometry and uh let's hope this lesson benefited you right so from meola most students favorite exam coach and the crew behind the scene, my team at Unlockwise.

[01:21:16]Uh we urge you to subscribe, comment, like, share and enjoy the experience as we are here to change the way people view education. Right? So uh until our next lesson is bye for now.