The formula to compute via hat looked kind of complicated. Is it difficult to work with in an explicit example? Well, only one way to find out that is try. So here we have an example. A w span of two vectors. You want a q2 explicitly given. See, you want a u2 or a tonal to each other. So they form an orthogonal basis of w. And I have some y to two six. And then the question is find y hat and find z. So how do we find y hat? We have an explicit formula. y hat equals projection on u1 plus the projection on u2. Projection on u1, u1 in y equals 2 plus 2 plus 4. Over u1 in u1, 1 plus 1 equals 2. times q1, 1 on 0 over there. And then u2 in y2 minus 2 plus 6 equals 6 over u2 in u2, 1 plus 1 plus 1 equals 3 times 1 minus 1, 1. So we have 2 over there, 2 plus 2 equals 4, 2 minus 2 equals 0. And 0 plus 2 equals 2. So there we have our y hat. Now, let us find z2. We know y equals y hat plus z. So that means that z equals y minus y hat. We have already computed my hat. We know z. So we get 2, 2, 6 minus 4, 0, 2. It shields minus 2, 2, and a 4. I remember, do not changes into 1 minus 1, 1, 2, or whatever. That's not the same. z equals minus 2, 2, 4. Let's check whether the answer is correct to be completely sure. So y hat plus z has to equal y. So let's see y hat and z. We get 2 plus y equals 6 equals y. So that's indeed correct. But we very stupid if it would be wrong, by the way, because we computed z like y minus y hat. And then z hat has to be 0, have to be orthogonal to each other. We get minus 8 plus 0 plus 8 equals 0. So that's fine. OK. So let's check. How can we use this, for example? It's the interpretation. Well, interpretation, between quotes, it's quite hard to draw the plane through you. One and you two, correct? These are just pretend that it was the x, y plane, just to make the sketch a bit easier. And then the y is somewhere over there. The y hat is the factor in the plane closest to y, to 0.p prime. So we can, for example, use the y hat to compute the distance from w to y. It's the same as the distance from p to p prime, which is the same as the length of z, c. We have z, the length of z equals 16 plus 4 plus 4, 24. So the distance equals the square root of 24. So that's how you can use this.