Let's talk about vectors. Vectors are one of the most important ideas in physics. They are measurements or variables that have both magnitude, basically how big the number is, how big the measurement is, and direction, whereas scalars only have magnitude.
Some examples of scalars are mass, speed, distance, temperature. These don't really have direction. Vectors, on the other hand, some examples would be weight, any force for that matter, velocity, which is just speed with a direction, displacement, which is just distance with a direction, acceleration as well, and there are other ones as well.
Basically, vectors are anything that you can represent with an arrow. The direction of the arrow gives you the direction of the vector, and the size of the arrow indicates how big the number is, what the magnitude is. And what we can do is actually add vectors together. So for example, say I have a displacement of 10 meters north, then I go 20 meters east. Adding these vectors together tells me basically where I've ended up compared to where I started.
If I was to draw an arrow going from beginning to end as the crow flies, that is my resultant vector. I could say in this case, my resultant displacement. What's the overall vector? One vector that represents all the other ones added up together.
If these component vectors are at right angles to each other, then I can use Pythagoras to find out what the resultant is, the hypotenuse of the right angle triangle. Hopefully you know how to do that. You should also know how to find the angle.
I'm gonna call that angle theta there. I'm just gonna say that's the angle of the resultant vector to the horizontal. What do I have?
I have the adjacent side and I have the opposite side. So therefore I'm going to use tan. Tan theta is equal to the opposite over adjacent. So that's 10 over 20 in this case. And then I can just inverse tan that to find the answer.
Let's say that I have here a ball that is being pushed upwards with a force of three Newtons. It's being pushed to the right with four Newtons. That's being also pushed this way with a force of five Newtons.
What's the resultant force now? Well, all I have to do is add them up, top and tail them. So here's my three Newton vector there. I'm going to add my four Newton one onto the end right there and then I'm gonna add my five Newton one onto the end there.
Lo and behold, we've actually ended up back where we started. And so what's the resultant vector going to be in this case? Zero.
There's no overall force on this bore. I could say that's Newton's first law then, couldn't we? The ball isn't going to accelerate. But that's really important, is that balanced forces always make a closed loop.
And that actually goes for any vector as well. If you have displacements and you add them up and you're back where you started, then you literally are back where you started. But let's get onto the slightly more tricky thing, the thing that we do lots of, especially in A-level physics, resolving vectors. In other words, using an angle to find a component or resultant vector.
So let's say I have a ship and it's travelling north, but actually there's a crosswind as well. So instead of going straight north, its velocity is actually in this direction here. And I'm going to say that's, let's say 20 metres per second.
However, I wanna know how fast is it going north? Now I know that its overall speed is 20 metres per second. Of course it's going north and also there's a crosswind pulling it to the right there.
So there are my component velocities. So I want to find a component. And components, well it's in the name, they're always smaller. Now in maths, you talk about trig and you talk about SOHCAHTOA, et cetera, but I find quite often they don't really tell you what sine and cos are. We'll forget about tan for now.
Sine and cos, they're just ratios between zero and one. That's all they are. They have a minimum value of zero and maximum value of one.
So basically all they do is tell you how much smaller something is compared to something else. So look at this. To find my component, I'm gonna call that v. I can say that v is equal to 20, but I know it's going to be smaller than 20. So therefore I'm going to times by, well, either sine or cos of this angle theta. Let's say that that's something like 30 degrees. So what do I do?
Well, yes, I could draw a triangle and then I could figure out how the adjacent and the hypotenuse, but that takes quite a bit of time. And sometimes it's not immediately obvious where the triangle is going to be. In order to save time, there's a really, really easy trick.
to help you resolve vectors quickly. Lots of people say, oh no, I prefer triangles. Trust me, once you start doing this, you won't stop.
So I can see that the angle of 30 degrees is between my resultant vector and the components I'm looking for. So therefore I could say that I'm sweeping through the angle to get from one to the other. Basically, the angle is between the two.
So therefore I'm going to times by cos of 30. We use cos if the angle is between our resultant and component vectors. And that gives me, 17.3 metres per second. Does that make sense?
Yes, it does. It's smaller. We know that whatever the component is of the velocity going north, well, it can't be faster than that 20 metres per second, can it?
What about if I wanted to find this horizontal speed here? Tell you what, I'm gonna call that VH. Call that VV for vertical. Well, it's gonna be similar.
I'm gonna start off with my resultant vector, and I know it has to be smaller than that, so I'm gonna times by sine, of course, but this time, we're timesing by sine of this angle. Why is that? Well, it's because the angle is not in between.
Now, here's the way that I remember it. This is my mnemonic. To go from the resultant to this vector here, we're turning away from the angle. In the Bible, you'll see that it says, turn away from your sin. If you're turning away from the angle to get to your component, which we are here, you use sine.
If you're turning through the angle to get from one to the other, then you use cos instead. And that gives us just 10 meters per second. So there we go. We've just found the horizontal and vertical components of this resultant vector, this resultant velocity. What about if we're asked to find a resultant instead?
Let's say that we have a little dude here. Let's call him Bob. and he wants to go for a walk eastwards 200 meters. However, Bob doesn't have a compass on him and actually he's ended up walking not due east.
He has gone 200 meters east, but because he's walked at a bit of an angle, he's actually gone further than that 200 meters, isn't he? So how would I find this distance here? So Bob checks his compass. He sees that he's walked at an angle of 20 degrees and he wants to know how far he's walked. But we know he's traveled further.
So therefore we're not gonna times by this ratio between zero and one, we're going to divide by this ratio instead. Are we gonna use cos or sine? Well, let's remember our rule, turn away from your sine. We're not turning away, the angle is between in this case, so we're gonna use cos instead.
So divide by cos 20, and that will give you a bigger number than 200. So it turns out that he's walked a little bit further, 213 meters. So there we go. Trust me, this trick is worth learning.
I haven't really drawn a triangle to resolve vectors in years and years, and it saves you so much time when time is precious in your exams. Let's have a quick look at a couple of examples real quickly. Think about work done.
Work done, the equation W, ugh, I prefer E, let's go with that, it is energy, is equal to force times distance, but that's only if they are parallel. So therefore, if we have a box like this, and there's somebody pulling it with a string like this, That's where the force is. But of course, it's only going to go horizontally. And we have an angle between theta.
Therefore, it's not the whole of the force that's being used to do work. No, in fact, it's only this horizontal component of the force. I can call that FH maybe.
That is the component of the force that is parallel to the distance. So therefore, if we wanted a proper version of this equation, it wouldn't be E equals FD. It would actually be F. times cos theta, and then we multiply that by the distance.
Or in other words, FD times cos theta. And that's probably how it looks in your formula sheet. That cos theta is there just to make sure that the components of the force is in the same direction as the distance.
Another example would be moments. Let's say we have a seesaw like this. Let's say there's somebody pushing down on this side of the seesaw with a force like that.
Now we know that a moment is equal to Again, force times distance, but we know that the definition of moment is force times distance perpendicular to the force's line of action. And so therefore, actually, we need to make sure that these are perpendicular. So it's the opposite to work done.
That's why it's the same variables, just not the same equation, strictly speaking. And there's the angle there. So I'll tell you what, let's pop in some numbers. Let's say that this box has a weight of 10. and it is a distance d away from the pivot. And so is this force here.
So what do we know according to our moments? Well, we know that this force here must be supplying a downward force of 10 Newtons as well. But what if you were asked to find what the force actually is? Well, it's gonna be more than that 10 Newtons, isn't it?
So let's pop a number in there. Let's call that 40 degrees. So we know that this resultant force here is equal to the 10 Newtons but we know it's bigger than the 10 newtons, so we're gonna divide by, what are we dividing by?
Well, the angle is between the resultant and the component, so it's gonna be divided by cos of 40, and that gives us 13 newtons. So we've just found a resultant from a component there. So there we go.
This comes up a lot in forces, of course, so if you wanna see more about resolving vectors, have a look at my balanced forces video. Hope you found this helpful. If you did, please leave a like. And again, if you want to see this principle applied, have a look at my past paper walkthroughs.
Thanks for watching. See you next time.