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Compute the quantities in Exercises $1-8$ using the vectors$$\mathbf{u}=\left[\begin{array}{r}{-1} \\ {2}\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}{4} \\ {6}\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}{3} \\ {-1} \\ {-5}\end{array}\right], \quad \mathbf{x}=\left[\begin{array}{r}{6} \\ {-2} \\ {3}\end{array}\right]$$$\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}}\right) \mathbf{v}$

$\left[\begin{array}{c}{\frac{8}{13}} \\ {\frac{12}{13}}\end{array}\right]$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 1

Inner Product, Length, and Orthogonality

Vectors

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Hello. We're going to continue working with dot products, so the quantity of interest this time is going to be given by the dot product You times you dot p divided by V v times the vector V and we'll just give the definitions. Here we have you is equal to it. Negative 12 and V is equal to 46 These are both two d vectors. And so you might be asking, What is this quantity you dot divided by V v times via. So this first term in the Prentice's is ah, ratio of dot products. This will be a scaler divided by a scaler which gives you a scaler. And the second quantity here. This V is a vector, and so will we end up with is scaler times a vector which gives us a vector so we can expect a factor in our final result on So now let's actually go through and derive this. So we have UW divided by V dive. Whoever multiply that by V. So, um, this is gonna be equal to you dot v is going to be negative one to ah, well, read it alphabetically. First, If we remember the DOT product is, uh, you don't b is equal toe u transpose v and feed obviously goingto be transposed fee. That was V. And then this is equal to negative one to transpose times 46 and then be transposed. V is going to be equal to 46 transpose for six and then times 46 of the output. So now we can evaluate these. So this will be a negative 12 and then for 46 and then 46 times 46 And now that the dimensions airline, we can apply the that product so we'll have this is equal to negative. One times four plus two times six and the numerator and then on the bottom will have four squared four times four a six squared six top six. And this will give us, um oh, and then 46 here on the up. And this will give us, um, negative floor plus 12/16 plus 36. I'm 46 of the output and then this gives us eight divided by 52 times the vector for six, which gives us to over 13 times for six or 8/13 and then 12/13. And this is our final output, which, as we said before, is a vector. Thank you.

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