Vectors 2025 paper 2 exam question

追加: 2026-05-12

[00:00:00]Hi everybody, welcome to this platform. This is samba.

[00:00:08]All right, so we've got uh this important question right here. If you can see nicely, we are talking about vectors.

[00:00:18]So when dealing with vectors, there are important things that you must must consider.

[00:00:27]Of course, direction is one of the key elements that you must consider when dealing with vectors. So, this is direction.

[00:00:42]So this direction it says that or before I say what the direction says or what you need to consider under direction you first need to identify the arrows.

[00:01:01]Arrows are very important and after identify the arrows you also get to know about the letters.

[00:01:12]So these are the two key elements when it comes to vectors to do with direction so to say.

[00:01:23]So now the arrows how do they work out?

[00:01:29]So for example when you see an arrow like this it's pointing in this direction and here there is a while here there's b this simply means the vector is coming from point A to point B and how we denote this we can denote it using either letter A or letter B anything Provided that provided that there is a there is a a bar down here.

[00:02:04]This simply means you're dealing with a vector.

[00:02:10]So if I were to ask you to find vector A.

[00:02:16]So there must be um a bar or an arrow on top here to show that this is a vector.

[00:02:23]Okay. So vector A this simply means we are talking about the A which is right up here that's vector A and um I talked about the arrows. So these arrows if they are pointing this normal direction you just get what they've given you here like it is. Now in a case where I've got something that looks like this or they ask you to find vector B a this is where now the issue of including a negative sign comes in because this vector is coming from A to B but the arrow or what we've been asked here the arrow is pointing in this direction starting from B to A. So it's the opposite. Now to show the opposite under vectors, we use negative signs.

[00:03:24]Let me talk about the letter. So now let's say you've been given a vector like this A B. There is no arrow here. They just tell you that if vector A is equal to B, you know that right up here it's a B.

[00:03:50]And maybe they say find find vector B A. Find vector B A. So what are you going to do here? You just consider this cuz these two letters are involved right here and they pointing in the what? In this direction. So now for you to show the vector here you are going to put a negative and then write the B because we are starting with B ending with A. Why? What we are given here we are starting with A ending with B. So this must highly be considered.

[00:04:33]So the next thing that we need to know when it comes to vectors is of course the ratios.

[00:04:42]Ratios are very important.

[00:04:46]So we have to know the ratios.

[00:04:50]So these ratios I'll just give you an example. these ratios.

[00:04:56]Let's say we've got uh a vector that looks like this. Here we've got a b and c.

[00:05:06]So now if they say a b 2 b c is equal to 1 2 3.

[00:05:22]So what you're going to do here is to just follow respectively AB is being represented using a one this one while BC it's a a three.

[00:05:35]So this is what we need to know. Now let's do this. Let's say we've got the value of this vector from A to I mean to C. So vector a c let's assume that vector a c is equal to a + b.

[00:05:57]Okay. So from here all the way to here it's a + b. Now if I ask you to find okay I ask you to find vector a.

[00:06:12]So find vector a. This is a ratio. It's not a vector. It's a ratio. So now we want to find the vector a in that we've been told that vector a c from a to c is equal to a + b. Now how do we get to find this one a b. So for this one what we're going to do here is to say one this one is for a b over the total which is a four. And where is this four coming from? You just add these two numbers.

[00:06:49]Okay, the ratios. So that is a 1 + 3 which will give us a four. And then you say of what? Of a c. So this vector a is a one over 4 over all this. Okay.

[00:07:08]All of this from a to c. That's why I'm putting a c here. So this will be a 1 / 4. What is our AC c? According to what we said here, we said AC is equal to A minus I mean A + B like this. You can even simplify like that like this. Or if you want you can say a + b that's a vector over 4. It's one and the same. So this is how we do this. Now let me show you again if we to find BC. So BC vector BC since we using the ratio will simply be equal to 3 over the total. We know that from here to here it's a three. So three over the total which is a four of a c.

[00:08:08]Okay. And we know of course that a c is equal to a + b. Then here this will be a 3 a / 4 + 3 b / 4 which is equal to 3 a + 3 b over 4 just like that. So this is what I wanted you to to get an idea of and then from this point let's do this from this stage the other important thing that I would like you to know is to know that if you've got uh let's say you've got this and then here it's a b See? Okay. We've got arrows here, arrows here, and there.

[00:09:13]So, if this is A and this is B. Under vectors, if I ask you to find vector A, you always start from pointy A or should I say the point which is starting. In this case, A is the one which is starting. So it will be a c plus c b. Have you seen? So we want to find vector a b. We'll move from a to c c to b. That's how it is. Okay. And we can indicate that this is our a minus b because a the arrow is pointing the normal direction which is a c from a to c. But for this one the arrow is coming from B to C. Here we've got the opposite which is a CP. So we have to indicate here or to put the negative sign. So that's how this one supposed to be done.

[00:10:15]Now let's try to apply these ideas which we've got in our question here. So the first question here they are saying in the diagram a d that's a vector vector a d is = 3 a vector a b = 2 b and a b to b c is = 1 to 2 that's the ratio and then bx to bg is = 1 to 4 okay so that's b X to B D is equal to what? To 4 and E is the midpoint. So E where is E? E right here is the midpoint over C D.

[00:11:06]So now here to do our calculations here or to work out they're saying express in terms of A and B uh A and O B vector BD. So B D the vector that we're talking about is what we are looking for B D and according to what we said we said when finding the vector you at point B going to D you are supposed to go or start from the same B go the other direction so go to A so from B to A plus A to G. So I know somebody may be asking that why didn't we go to C?

[00:11:55]This side we don't have the the the vectors to use here. The reason I've used this side because we've got a 2 A here and a three I mean 2 B and a 3 A.

[00:12:06]So we can plug in what is our BA? It's a -2 B. Remember the arrow is pointing in the direction of B from A to B. But what we want here is from B to A. We have to indicate the the negative negative sign.

[00:12:26]And then for A D it's a normal direction. So we just write 3 A. Then here we can rewrite this. Start with negative value. I mean we start with a positive value which is a 3 A which has got a positive and then 2 B follows. And this is our answer for question one. We go to question two. So they want us to find d x.

[00:12:58]So for vector dx, d is right here. x is here. Remember there is a ratio involved. So let me go or let me take you back to the ratio. So the ratio which is involved here it's this one of course.

[00:13:17]B X to B. So if you're able to see B D it's a it's a it's a total. So this is one bx it's one and then B D it's a four. So we're not going to write a four because this is a total from here to here. We therefore subtract one from four it will give us a three. So this is a three such that if we were to add these they will give us the total of the ratios from here to here. Okay. So now we've known that. So what we can do here we can say dx which we are looking for this one dx okay is the negative okay 3 over the total which is a four. I'll explain why I've put the negative. So the reason why I've put a negative here is because what we are looking for is starting from D going to X. But the ratios that we have they are starting from B going in this direction. Okay, going to the D direction.

[00:14:29]And for us to indicate here we're supposed to put a negative to show that it's the opposite direction. So we've written 3 over the total which is here over b d over b d there is there is already a negative sign which shows that it's uh it's a it's moving in the opposite direction. So this of course will give us a -3 over 4 open bracket.

[00:15:03]This vector bg is what we just found here. That is a 3 a - 2 b. Then we can simplify. Of course, this will give us a a 9 a / 4 + 6 b / 4 just like that. So if you want we can even change we can say this of course will give us a 3 over 2 b. So 2 into 6 it's a 3 into four it's a two and then you say minus 9 a over 4. So this is the answer for question two.

[00:15:52]We now go to question question three where they want us to find the vector C D. So let's go back to the diagram.

[00:16:03]C is right here. Okay. And then D is right over here. So they want us to find C D. Now in order for us to find CD we again need to use the ratio because for CD to be found we supposed to move from C to A A to D. So this will be C A plus A D. Okay that's it. So what we're going to do here following the ratios here they've given us that AB to BC is equal to 1 to two. So this is one and this is a two. Okay? Have you seen this is one from A to B is a one and then B to C it's a two following the ratios here. So what we're going to do here we're going to multiply. Okay. So this from here to here it's a 2 B.

[00:17:10]Meaning from here to here it's a 2 * 2 B. So this multiplied by 2 B. So there are two of the 2 B from here to here following the ratio. And then what? And and and and at this point from A to B there is only one of this which is a one. So that's why we multiply and we are and this is giving us a 4 B or it has given us a 4 B. So we now know that from here to here the vector is a 4 B while from here to here it's a 2 A. Now the total from here to here which is AC c. So our AC c will simply be equal to a 6 b just like that. And this is where we want AC. A C we know that it's a 6 C B plus. So when we check nicely here we need to put a negative we have to put a negative sign. The reason why we are putting a negative sign we know that AC c is equal to what? It's equal to 6 b. But what we have here it's opposite which is a c a. So we have to put a negative sign while our a d is a 3 a. So this 3 a three uh a from a to d. So now what we can do here is just to say 3 a - 6 b is our answer. We go to the next question. So the next question here they want us to find a e.

[00:18:48]So a e when we check nicely from a to e it will be equal to a g e or if you want you can say a c c e. So let me use a c plus c e. So what is our ac? Our ac we just found here. It will be equal to 6 b plus our c e. You see c e. We can use the previous answer here because we found cd and they're saying this is a midp point. So we can just divide our a d by 2 so that we get this part which is a a c e. So we can say 3 a - 6 b over a 2. Then this will be a 2. This will give us a 12 b.

[00:20:00]And then here + 3 a - 6 b. And when we simplify together this and this, you see 12 b - 6 b will just give us a 6 b + 3 a. Okay. So this is what we're getting over a 2. We can factor out a three if you want. So 3 over 2, we shall remain with a 3 b + a. And this is our answer.

[00:20:34]Thank you so much for watching.

[00:20:35]Hopefully you picked one or two things.