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In Exercises $3-6,$ verify that $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ is an orthogonal set, and then find the orthogonal projection of $\mathbf{y}$ onto $\operatorname{Span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$$$\mathbf{y}=\left[\begin{array}{r}{6} \\ {3} \\ {-2}\end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l}{3} \\ {4} \\ {0}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}{-4} \\ {3} \\ {0}\end{array}\right]$$

$\left[\begin{array}{l}{6} \\ {3} \\ {0}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 3

Orthogonal Projections

Vectors

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first, let's make sure you want to need to our or thought so we're just gonna take without product. So we get lying is 12 plus 12 plus zero which in fact gives the zero. So these are thought of each other. So now we want to find the orthogonal projection off. Why? On to the span of wanting you to s? So we have a formula for this. So on vector, Why hat? Um it's equal to the projection do you want? Which is why one you wonder if one time they want and the projection of why onto you too. So we take the dot product with you too. You too dot product with itself times you too. So let's go ahead and find these coefficients off to the side. So why I got you. One is equal to 18 plus 12 plus zero. So we get 30. You won dot product with itself. You're square all the components and I'm together. So we have nine put 16 which gives us 25 now for the 2nd 1 we have. Why dot You too negative. Four times six is minus 24. Three times there is mine. Zero so negative. 24 plus nine gives us negative 15. Lastly, YouTube out product with itself, you get 16 plus nine, which is 25. So now I can fill in these coefficients. Ah, so this 1st 1 is a 30 over 25 and the second is negative. 15 over 25 so we can simplify a little bit, get six months and minus 35th. So now let's go ahead and play. He's out 6/5 times. You won so 6/5 times three is 18 over five, 65 times four is 24 or five and then we get a zero. No, we have negative 3/5 times. You too. So negative. 3 50 times negative for is 12 over five negative, 3/5 times Theories negative. Nine over five and then a zero. So adding is together 18 plus 12 gives us 30 divided by +56 24 minus nine is 15 divided by five is three and 0.0 s. Our final answer is the Vector 630

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