00:01
So in this question we have y, which is 5 minus 2, 1.
00:08
And we want to find the orthogonal projection onto the span of u1 and u2, where u1 is minus 1 -1 -0, and u2.
00:26
And u2 is 2 -4.
00:30
So essentially what we need to do is to subtract off the part of y, which is orthogonal to both u1 and u2.
00:39
So the orthogonal projection, yp, is going to be y minus y orthogonal.
00:48
But how are we going to calculate y orthogonal? well, y orthogonal.
00:53
U1 equals y orthogonal .u2 equals 0 .u2 equals 0.
01:00
And since we're in three dimensions, if there's a vector which is orthogonal to u1 and u2, then y orthogonal must be parallel to it.
01:10
And such a vector does exist, u1 cross u2 is also orthogonal to u1 and u2 so in 3d y orthogonal y -orthognal must be parallel to u1 cross u2 so we can write that y projected is y minus y dot u1 cross u2 divided by mod u1 cross u2 because that's the projection of y onto this orthogonal unit vector.
01:54
We've divided out this so that it is indeed a unit vector.
02:01
Okay, so now we need to...
02:07
Oh, and this needs to be in the direction of u1 and u2 as well.
02:11
So u1 cross u2 over mod u1 cross u2, so that that's now a vector.
02:22
Right.
02:23
And so basically this is saying, how much of y is orthogonal, and this is saying what is that orthogonal direction.
02:30
So now we need to calculate a few things.
02:33
Let's calculate u1 cross u2 as a unit vector.
02:37
So let's say that o hat, actually let's say t hat is u1 cross u2 over mod u1 cross u2.
02:50
And this is going to be minus 110 crossed with 224 divided by the modulus of that.
02:58
So what's this going to be? so the x component we do 4 minus nothing, so that's 4.
03:13
The y component we have nothing minus minus 4, so that's also 4.
03:19
And the z component we have minus 2, minus 2, so that's 0.
03:25
And we need to divide by the square root of 32, so we divide by 4 root 2.
03:31
So t is 1 over root 2, 1 over root 2, 0.
03:42
So that is our t, and now we can go into this formula here and write that y projected is y minus y.
03:58
.tt.
03:59
So that's going to be 5 minus 2 1 minus 5 minus 2 1, dotted with 1 over root 2, 1 over root 2, 1 over 2, 0 in the direction of 1 over root 2, 1 over root 2 0.
04:19
So now what do we get here? so 5 minus 2 1 minus.
04:23
Now let's dot these together...