Hey everyone, and welcome back. So in recent videos, we've been talking about operations you can do on vectors, and one of the operations we discussed was the product of a scalar and a vector. When multiplying a vector by a scalar, what this does is actually stretches or shrinks your vector in some kind of way, depending on what that scalar is, but the result you get is another vector. Now what we're gonna be discussing in this video is another operation called the dot product, and the dot product is actually a way to take a vector and multiply it by another vector. Now the results you get when you do a dot product are actually quite interesting. So without further ado, let's jump into some examples that you'll likely see in this course.

So let's say we have these 2 vectors, vector v and vector u, and we wish to find their dot product. Well, when doing a dot product, what you want to do is multiply the like components together, and then you want to add their results. So let's say we have these 2 vectors right here. What I can do is I can multiply the x components together and the y components together, and then add them all up. So let's go ahead and try that with the 2 vectors we have here. So if I wish to find the dot product of v and u, what I can first do is multiply their x components. And their x components, I can see are 1 and 3. So we're gonna have 1 times 3, and we're going to add this to the product of their y components, which is going to be 2 times 2. Now one times 3 is equal to 3. 2 times 2 is equal to 4, and 3 plus 4 is 7. So the dot product and the solution to this problem is 7. So this is how you can find a dot product. It's a pretty straightforward process, but I want you to notice something interesting. Notice when we took the product of these two vectors, we ended up getting a scalar, and that's the interesting result I was discussing before. Now, what this number tells you is this number generally represents how close the vectors are to pointing in the same direction. So a higher number would mean that the vectors are more closely aligned.

Now to make sure we really understand this concept and operation, let's actually try a few more examples. Now in this problem, we are asked to complete the following vector operations below, and we're going to start with example a. Now example a asks us to find the dot product of vectors u and w. Now I can see that vector u and w are given to us on this graph, so we can just go ahead and find their dot product. Now vector u, I can see is 3 I, and 3 I is the same thing as 3 I plus 0 j. All this 0 j tells us is that we have no direction in the y direction because this vector is completely along the x axis. And what I'm going to do is find this dot product with vector w, which is 2 I plus j. Now when doing this operation, all we need to do is multiply the like components and add the result. So we're going to have 3 times 2, which is 6, and that's going to be added to 0 times 1 which is 0. So we're gonna have 6 plus 0 which is 6. And this right here is the result for our dot product. Now what exactly does this number mean? Well, it tells us how aligned vectors w and u are. And since I can see that the vectors actually are close to pointing in the same direction, we can see that they do have alignment, so we get a positive result, and that is non zero. But now let's go ahead and try example b.

Example b asked us to find vector v dotted with vector w. Now I can see that we have these two vectors right here, so let's go ahead and find their dot product. Vector v, I can see is negative 2 I plus 3 j that vector is right here, and I can dot this with vector w, which is 2 I plus j. That can multiply the like components, so we'll have negative 2 times 2, which is negative 4, and I'm going to add this to 3 times 1, which is 3. And negative 4 plus 3 is negative 1. So this right here is the result for our dot product. But what exactly does it mean when we get a negative result in the dot product? Well, what this tells me is we have negative alignment. That means the 2 vectors are not aligned at all, they're actually pulling against each other. And since it appears that these two vectors are actually pulling opposite ways, it makes sense that we got a negative result for our dot product.

But now, let's try example c. An example c is a bit more complicated because you can see that we have vector u, but this is going to be dotted with vector w plus v. Now my first step here is going to be to figure out what w plus v is. I can see vector w is 2 I plus j and I can see that vector v is negative 2 I plus 3 j. So all I need to do is add these vectors together to get this result. Now negative 2 plus 2, that's going to be 0. So we have 0 in the I direction and then I'll add this to 3j plus j which is 4j. So this is the result that we get for vector w plus vector v, and what I need to do is find this dot product which is vector u, which you can see here is 3i+0j and this is going to be dotted with vector w, which is 0i+4j. Now what I can do is multiply the like components and add the results. So we're going to have 3 times 0, which is 0, plus 0 times 1, which is 0. And 0 plus 0 is 0, so that's what we get for our dot product.

Now we've talked about what a positive result means and what a negative result means, but what does a 0 result mean? Well, a 0 means we have no alignment whatsoever. These vectors do not pull on each other, nor do they point in the same direction in any way. They're completely not aligned. And so what this means is if we have a dot product where the result is 0, the only way 2 vectors could not be aligned at all is if they were perpendicular. Now if I were to draw a vector 4 j on our graph, it would look something like this. It would point in the y direction, and notice how these two vectors are perpendicular. And because these vectors are perpendicular, that means their dot product is 0. So that's what it means to get a 0 dot product, a negative dot product, and a positive dot product. So, I hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.