Introduction to Vector Math

by Patrick Ford
386 views
3
2
Was this helpful ?
3
Hey, guys, We're gonna be working with vectors a lot in physics, so you're gonna have to get very good at how we manipulate and combine them. So in this video, I want to introduce to you what vector math is and what it's all about. But I want to take a minute to describe what the next few videos they're gonna be about. We're gonna be working with vectors with all these diagrams with these little axes and these grids and boxes because it's a great way to understand, very visually what's going on vectors. Then, later on, what we're gonna see is we're gonna see all of the math equations that describe vectors. So let's get to it. So adding and subtracting scale er's is pretty easy to remember. Scaler Zehr Just simple numbers, But vectors have direction. So Mathis sometimes not a straightforward. For example, if you were to combine scaler, let's say you're moving and you're combining a 3 kg in 4 kg box, Then the way that you add these boxes together, if you were a lump them together in a single box is you just add three and four straight up, so three plus four is just seven. So combining scaler, XYZ just simple addition where things get a little bit more tricky is when you start combining vectors and it's really just two cases you're going to see you combine parallel vectors. Then there would be like walking 3 m to the right in 4 m to the right. So because vectors have direction, they're drawn as arrows. So this 3 m to the right and 4 m to the right would look like this. So this is 3 m and this is 4 m. Now, if you were to combine these two vectors together, it would basically just be as if you walked 7 m. So when you combine parallel vectors, the total displacement, if you will, is three plus four, which is 7 m. And so when they're parallel, they just add together like normal numbers. By the way, the same thing would work. If you actually had these two vectors together, let's say you had three and then four like this. Well, that would also just form a total displacement of seven. So as long as these things are parallel in whatever direction will always add together where things get a little bit. Trickier is when you're combining perpendicular vectors. So now let's say I walk 3 m to the right and then 4 m up. So we got this grid that's gonna help us visualize what's going on here. So I only walked 3 m to the right. So 123 and then I'm gonna walk 4 m up like this. So this is four, and this is three. Now, you might think the total displacement is just three plus four, which is seven. But it's actually not because your total displacement is really just the shortest path from where you start to where you end. So you were three and four. You're actually starting over here and you're ending over here. So what happens is we make a little triangle like this. So this is my total displacement. So how do I calculate this? Well, I can't just use simple addition. I'm gonna have to use a different kind of math. And this, actually, if you'll notice here, makes just a triangle. So vectors just make a bunch of triangles. The way we saw for this is by using an old idea an old math equation from algebra or trigonometry called the Pythagorean Theorem. So this is the Pythagorean theorem, which says that a squared plus B squared equals C squared. So as long as you know, two sides of a triangle, you can always figure out the third one. So I'm gonna call this a this one b, and this one is C. So if I want to figure out the sea, all I have to do is just c equals the square roots of a squared plus B squared. So that means my total displacement is this square roots of three squared in four squared, and that's actually equal to 5 m. So even though I went three and four, this actually is five in this direction and basically so we can see is that vectors just form a bunch of triangles. So that means that the way that we're going to solve vectors is just by using a lot of triangle math. That's really all there is to vector math. Let's go ahead and get some more examples. So I'm gonna walk 10 m to the right and then 6 m to the left. So I'm gonna walk 10 m to the rights so there's gonna be my 10 m and then m to the left. So something like this. So my total displacement, where to draw the displacement vectors which we are did and calculate the total displacement while if I walked 10 to the rights and then six to the left, then my total displacement is actually justify started here, and I ended here. And so this is just 10 minus six, which is just 4 m. So that's pretty straightforward. It's because these things air parallel or technically, they're anti parallel, but they sort of all long lie along the same plane or the same line like this is just Some of them are forwards and backwards. So now we're gonna walk 6 m to the right and 6 m and 8 m down for part B. So now I'm gonna walk 6 m to the right. Let's say it's like that, and then 8 m like this doesn't have to necessarily be to scale. So what is the total displacement? Well, just like before now we have perpendicular vectors, so we're gonna draw this line here that connects these two points, and this is going to be a triangle so notice how it just forms a triangle like this. And I wanna I wanna figure out with this, uh, see, or the high pot news of this triangle is so I'm just gonna use the Pythagorean theorem, so C is equal to six squared plus eight squared. Notice how you don't have to plug in. You know, you have to worry about signs or anything like that. So you're just gonna do six and eight and this equals 10 m. So this is the displacement over here. It's 10. Alright, guys, that's it for this one. Let me know if you have any questions.