INSTRUCTOR: Express the vector v equal to 5i minus j as a sum of orthogonal vectors such that one of the vectors has the same direction as the vector u equal to 4i plus 2j. To work with this question, we need to follow a couple of steps. First is we need to find the projection of v onto u. That's going to guarantee that we're in the same direction as the vector u. That's just how that projection is defined.

So I have the definition written here. And I'm going to call this vector p. Vector p equals projection of v onto u equal to u dot v divided by the magnitude of u squared. Then that's a scalar value multiplied by the vector u. So that takes care of the same direction.

Two, we need to make sure that these two sum to be the vector v. Well, to do that, we're going to have the vector q equal to the vector v minus the vector p, that projection that we just found, and subtract it from v to guarantee that the two vectors we find sum to v.

And then finally, we want to make sure that they are orthogonal vectors. So we'll check for orthogonality. To do this, we'll find the dot product of p and q, those two vectors, and if, in fact, they are orthogonal, then their dot product will be equal to 0.

So let's begin step 1, refining our projection, vector p. And based on that definition, we need to find u dot v as well as the magnitude of u. Let me just write over here, in component form, u equals-- again, in component form, this would be 4 comma 2 and v will be equal to vector 5 comma negative 1.

So u dot v will be equal to-- because we multiply component-wise and then sum those results for our dot product, this will be 4 multiplied by 5 plus 2 multiplied by negative 1. So that'll be 20 minus 2, so that'll be 18. So the dot product is 18.

And then we also need the magnitude of u squared. Now, by definition of magnitude, we will square the components, sum them, and take the square root. However, in this case, we're also going to square that result. So we'll just forget the square root part of this.

So we'll take the vector components 4 and 2. We'll have 4 squared plus 2 squared. Four squared is 16, 2 squared's 4, so this result is 20. Now, because it's going to be a scalar, let's go ahead and reduce that. We'll say 18 divided by 20. That'll be 9 over 10. So this'll be 9 over 10.

And then we multiply that scalar by the vector u, which is, again, 4, 2. Which means our first vector, once we multiply by that scalar, will be 3-- let's see. That'll be 36 over 10, which is 18 over 5. So our first component is 18 over 5. Our second, we'll say 9 over 10 multiplied by 2, and that will be 9 over 5.

All right. Second, we will take the vector v. And again, let's just find the second vector q. We'll take the vector v minus the vector p, that projection we just found. Sorry. I'm fixing my notation because for some reason, I keep wanting to switch back and forth. There's a couple ways to write vectors, and keep pointing to switch back and forth.

Actually, I did that over here on orthogonality testing, so let's fix that. OK. So if we subtract that, we'll take the vector v, which is 5 comma negative 1, and then subtract the vector 18 over 5 comma 9 over 5. Do that component-wise.

So 5 minus 18 over 5, and that is going to be 7 over 5. And negative 1 minus over 5 would be negative 14 over 5. So our second vector is first component, 7 over 5, second component, negative 14 over 5.

Finally, we want to make sure that these two vectors are orthogonal. So step three, we'll find the dot product of 18 over 5 comma 9 over 5 and 7 over 5 comma negative 14 over 5.

18 over 5 multiplied by 7 over 5. And then we have a negative there, so go ahead and do plus 9 over 5 multiplied by negative 14 over 5. And that is, in fact, equal to zero.

So the answer to this question is vector v equals the sum of two orthogonal vectors. Vector v equals vector 18 over 5 comma 9 over 5 closed vector plus the vector 7 over 5 comma negative 14 over 5.