Introduction to Dot Product (Scalar Product)

Hey, guys. So in previous videos, we saw how to multiply vectors by scaler, which is a simple numbers. Now it's pretty straightforward, but in some problems gonna have to multiply vectors by other vectors. And what's weird about Vector multiplication is that there's a lot of not intuitive things that are going on. It's hard to visualize, but I'm gonna show you that vector multiplication actually just boils down to a really simple equation. Let's check it out. So there's actually two different ways to multiply a vector together or multiply two vectors together. One of them is called the dot or the Scaler products, and the other one's called the Cross or the vector product get they're doing. They get their names from their notation. So if you have two vectors A and B, the dot product is gonna be with a big fat dot in between them and the cross product between two vectors A and B is gonna be a big X sign like this. And so there's a couple of differences between them. Um, the cross product is actually we're gonna be covering in a later video here, so for now, we're just gonna focus on the dot product. So how do we do a dot b. But before we get there, I actually want to cover something that we've previously covered before, which is how you multiply vectors by scale. Er's We call this multiples of vectors, for example. It was really straightforward. If you have a vector, that's three to the right. Sorry. Forward to the right and you multiply it by a number three. Then you just get another longer vector out of it. So it's basically is if you're just changing together, tip to tail these four vectors, these you know, these fours three times to the right, and you just get a bigger vector in this direction. That's a magnitude of 12 and it points in the same direction. The dot product is different, though, So when you do the dull products, you're actually gonna take a vector like this four to the rights and you're gonna multiply it by another vector, like three to the right and which weird about this scaler or the dot product is that you just end up with a number you're just gonna end up with 12 not the vector. So again, if you multiply this four vector by this three number you got at Vector. But for the dot product, you're gonna multiply a four vector by a three vector, and then you just get a number, which is 12. You actually don't get a vector 12 in this direction. It's not what you get. It is kind of strange. And so it turns out there's actually just a really simple equation that will help us do this dot product. So if you have dot product A and B, what you're gonna do is you're gonna do the magnitude of a which is written by the absolute value signs times the magnitude of B and then you're gonna multiply this by the cosine of the angle, where this angle is the smallest angle from A to B. So it's always this angle is drawn between a and B here, they're actually straight lines, but we're going to see how a bunch of different directions will affect this dot product. And so just make sure that your calculator is in degrees mode. Now, whenever you're doing the dot product, here is one way that I like to visualize or think about it. The dot product is basically when you're multiplying parallel components. That's how we're gonna think about this. Let's just do a bunch of examples. We can see how this how this equation works out and what it means. So we're gonna calculate the dot product between these two vectors right here. The first thing I want to point out is that any time you're calculating the dot product, you're always gonna line up your vectors so that their end to end so basically so that they start at the same point another way. Think about this tail to tail. So let's check it out. We've got these vectors A and B and trying to calculate the dot product. So a dot B is gonna be a B times the cosine of theta. So let's check it out. So we know the magnitude of eight times the magnitude of be Well, I've got these two vectors. I've got three times four, So I've got three, and then I've got four times the cosine of the angle, so I'm just gonna get 12 and then what's the cosine of the angle between them? Well, this one's at zero degrees, and this one's at zero degrees, so they're actually both completely parallel. So the cosine of zero. So these are parallel vectors. And when we think of think of the dot product, we're gonna multiply parallel components. Well, these vectors air perfectly parallel. So if you plug in CO sign of zero into your calculator, you're gonna get one, which means that our dot product is just 12. Let's move on to part B. Now we've got the same exact vectors three and four now is just one of them is raised to an angle. But let's just go through our equation. We've got a B or a dot B is gonna be a b cosine theta. So when we work this out, we already know A and B is gonna multiply to 12 right? These numbers don't change. The only thing that is changing, though, is the coastline of the angle between them. So now what's this angle? What's the smallest angle between A and B? Well, I've got this 10 and this one it's 60. So the gap are the difference is just 60 degrees. So that's when I plug into my calculator 60. And if you work this out, we're gonna get 12 times 0.5, Which means that my doubt product is six. So what's the difference here? What actually happened? Well, one way you can think about this is that this be vector here doesn't completely lie parallel to a but it does have a components that lies parallel to a So this is my ex component, my B X. And remember, we calculate this by be co sign of data. So one way you can think about this equation here is you're multiplying A which is three times be co sign of data. So if you do be co sign of data where you plug in. So if you have, this is four and this is 60 degrees, then my b X or my be cosign of fate. A term over here is gonna be four times the cosine of 60 which is gonna give me too. So basically, it's like we're multiplying this to buy this three, not the four by the three. So that's why we get a six out of it. So it's multiplication off parallel components. This three and these two are both parallel. And that multiplies to six. Alright, guys, let's keep going. So now we've got part C, which is we got the same vectors now be just points to the left. So we're gonna start off with same exact equation. We know a dot B is gonna be 12 times the co sign of theta. Right. So I've got 12 now. What's the cosine of the angle between them? Well, I've got these perfectly anti parallel lines like this, so that means that the angle between them is gonna be the two little quadrants like this. And that's just gonna be my fate is equal to 180 degrees. So if you plug in the co sign of 1 80 into your calculator, you're gonna get 12 times negative one. So that means our dot product is gonna be negative. 12. So what's this negative about? Well, all this really means is whenever you get a negative dot product, it just means that you're components point in completely or not, not completely, but it just means that they point in opposite directions. So here the parallel components to a is actually anti parallel. It's a special case of parallel, and so because they point in opposite directions it just picks up a negative sign. That's all there is to it. Let's move on. So we've got these vectors again. Three and four. So let's just go to the equation A dot B is equal to now. We've got 12 times the coastline of data. So you've got 12 times the coastline of what? What, what? What? We plug into our calculators. Is that the 60 degrees? Well, remember that this fate a angle has to be the smallest angle between A and B. So this is actually not the angle that we're going to use. We have to find out what this angle is. And this angle here is just gonna be 180 degrees minus the sixties. So our state is actually 1 20. So if you're gonna plug in 1 20 into your calculator, you're gonna get 12 times negative 120.5. So now our dot product is actually negative six. And so when we can think about this, is this be vector isn't completely anti parallel to a but it does have this component over here that does lie in the opposite direction to a So if you work this out this BX components, you would actually have to use this be co sign of data. You have to use the B 60 and you would end up with a component of negative, too. And so you're gonna multiply this negative to buy this three. And then that is what is what's gonna get you the negative six. So basically, it's those two multiplication of anti parallel components and you get the negative number there. All right, so for the last one, now we're gonna multiply this three and four. Start off with the equation a dot be is equal to 12 cosine theta. So that's equal to 12 times the cosine of what? Well, the smallest angle here is gonna be this one right here. And this actually forms a right angle, these air perfectly perpendicular like this. And so this angle here is 90 degrees. If you plug in 90 degrees, the coastline in nine degrees, you're just gonna get zero into your calculators. Doesn't matter how big these vectors are. Could be a billion. It could be a trillion. It doesn't matter, because the coastline of this angle here is zero. So that means that your dot product is equal to zero. And so what you can think about this is that the zero dot product just means that the components are in completely perpendicular directions. There is no components of B that is parallel to a because they're perfectly perpendicular. And by the way, this works for any combination of vectors. As long as you can figure out that the angle between them is 90. So even something like this, they don't necessarily have to be on the X and Y axis like this. As long as you can figure out that this angle is 90 the dot product will always be zero. Alright, guys, that's it for this one. Let me know if you have any questions.