>> It turns out that we can project this, this vector U onto a vector that is along that 20 degree angle. Consider this. We want to know how fast the object is traveling relative to an angle of 20 degrees. Here's what we know about the problem. Here's our vector U, and it's traveling at 50 feet per second along this angle of 60 degrees. All right. We know from our calculation over here that it's traveling horizontally at 25 feet per second, and it's traveling vertically at 25 root 3 feet per second. But that's not what we really want to know. What we really want to know is how fast that rascal is traveling at an angle of say 20 degrees for our discussion. So how fast is it traveling in this direction. That's what we're asking ourselves. So what we want to do is project this vector onto that line or that vector, you see, and, and, again we'll have a, a magnitude. The direction is, we know to be 20 degrees here, so that's what we're aiming to do. Now, let's understand that this vector, which I'll call V here for discussion, can be described in a number of way. I mean, since we know the angle is 20 degrees, we could describe this as a unit vector in unit vector form of, now remember R cosine Theta R sine Theta? Well, R is 1 here for unit vector. So it'd be 1 cosine 20 degrees 1 sine 20 degrees. Now realize that cosine 20 degrees is just a number in the calculator and sine 20 degrees is a number in the calculator. So it's just the combination of those two numbers that gives us the idea of a vector headed in a certain direction. Now, but, but it's the relationship between those two numbers that give it its direction. And if we can come up with a couple of other numbers, you see, that have a resulting direction of 20 degrees, any combination of such numbers would work fine. For example, I've calculated that, that if we use V to be 11 4 that we'll have the same 20 degree direction associated with this vector that is written in component form. All right, now I, I'm not trying to, to get us to calculate the 11 4. Sometimes it'll automatically be given to us. What I'm trying to point out here is that there may be a variety of ways of dealing with this vector that is going at this angle of 20 degrees. We really don't care so much about its magnitude. You know, we don't, we don't care about that. We can have, we can use as a relationship with our U vector a vector that has any magnitude along that 20 degree angle because our formula is going to allow us to make the calculation of magnitude along this vector associated with the U vector. You see, the object is traveling at 50 feet per second in this direction, but all we're trying to find out is how fast it's traveling in this direction. And we can use as a reference any value or any kind of notation for V. We don't care what its magnitude is. We're going to project this on to that, and it's just going to go a certain distance along that, that angle, you see. That's the, that's the idea. Okay. So here, the formula. The projection of U onto V, the notation is this. The projection of U onto V, you see, is, now just kind of look at the formula. It's a dot product of U and V in the numerator. It's the, the square of the magnitude of V in the denominator, and we're multiplying times V. Now what's so important here to, to realize is that this a scalar quantity. Now that means it's just, it's one real number to describe the value that comes out of this.