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Let $\mathbf{y}=\left[\begin{array}{l}{2} \\ {3}\end{array}\right]$ and $\mathbf{u}=\left[\begin{array}{r}{4} \\ {-7}\end{array}\right] .$ Write $\mathbf{y}$ as the sum of two orthogonal vectors, one in Span $\{\mathbf{u}\}$ and one orthogonal to $\mathbf{u} .$

$\hat{\mathbf{y}}=\left[\begin{array}{c}{-\frac{4}{5}} \\ {\frac{7}{5}}\end{array}\right]$$\mathbf{y - \hat { \mathbf { y } }}=\left[\begin{array}{c}{\frac{14}{5}} \\ {\frac{8}{5}}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 2

Orthogonal Sets

Vectors

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Lectures

02:56

In mathematics, a vector (…

06:36

02:00

Let $\mathbf{y}=\left[\beg…

01:45

Determine if the pair of v…

Find the projection of $\m…

05:45

In Exercises $25-28,$ find…

01:26

Calculating orthogonal pro…

00:26

In Exercises $23-24,$ prov…

00:46

Use a dot product to deter…

02:07

00:54

Find $a$ so that the vecto…

Okay, This question wants us to write the vector. Why, as some of vectors perpendicular and parallel to you, So we'll start by finding the parallel component. And that's the same thing. Is asking What's the component of why along you which we get straight from the projection formula. So we're projecting Why on to you? So that would be Why don't you divided by the magnitude of you squared distributed into you and the dot product would be two times for minus 21. So eight minus 21 and eight minus 21 is just negative 13. So that's negative. 13 The magnitude of you squared would be 16 plus 49 giving 65 and then our vector you is just for negative seven And this vector is the same thing as negative 1/5 times four negative seven. When we reduce, so are parallel Vector would just be negative. 4/5 and 7/5. And now, since we know that why is formed of parallel and perpendicular components, all we have to dio to find the perpendicular component is subtract this parallel component from the original factor. So our original vector Y is 23 and are parallel Vector is negative. 4. 57 50. So why perpendicular would be tu minus 4/5 giving us 14 5th and then three minus 7/5 which would be 8/5. And that is our perpendicular vector. So we have our two answers right here.

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