Difference between revisions of "031 Review Part 2, Problem 6"

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<span class="exam">(c) Let &nbsp;<math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L.</math>
 
<span class="exam">(c) Let &nbsp;<math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L.</math>
−
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 +
|-
 +
|'''1.''' The distance between the vectors &nbsp;<math style="vertical-align: 0px">\vec{u}</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; is
 +
|-
 +
|
 +
::<math>\text{dist}(\vec{u},\vec{v})=||\vec{u}-\vec{v}||.</math>
 +
|-
 +
|'''2.''' The orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\hat{y}=\text{proj}_L \vec{y}=\frac{\vec{y}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}\vec{v}.</math>
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|First, we calculate &nbsp;<math style="vertical-align: -4px">||\vec{v}||.</math>&nbsp;
 +
|-
 +
|We get
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{||\vec{v}||} & = & \displaystyle{\sqrt{(-1)^2+3^2+0^2}}\\
 +
&&\\
 +
& = & \displaystyle{\sqrt{1+9}}\\
 +
&&\\
 +
& = & \displaystyle{\sqrt{10}.}
 +
\end{array}</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Now, to get a unit vector in the direction of &nbsp;<math style="vertical-align: -4px">\vec{v},</math>&nbsp; we take the vector &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; and divide by &nbsp;<math style="vertical-align: -4px">||\vec{v}||.</math>
 +
|-
 +
|Hence, we get the vector
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\frac{1}{||\vec{v}||}\vec{v}} & = & \displaystyle{\frac{1}{\sqrt{10}}\begin{bmatrix}
 +
          -1 \\
 +
          3 \\
 +
          0
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          \frac{-1}{\sqrt{10}} \\
 +
          \frac{3}{\sqrt{10}} \\
 +
          0
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Using the formula in the Foundations section, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\text{dist}(\vec{v},\vec{y})} & = & \displaystyle{||\vec{v}-\vec{y}||}\\
 +
&&\\
 +
& = & \displaystyle{\Bigg|\Bigg|\begin{bmatrix}
 +
          -3 \\
 +
          3 \\
 +
          -5
 +
        \end{bmatrix}\Bigg|\Bigg|.}
 +
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|Continuing, we get
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\text{dist}(\vec{v},\vec{y}) } & = & \displaystyle{\sqrt{(-3)^2+3^2+(-5)^2}}\\
 +
&&\\
 +
& = & \displaystyle{\sqrt{9+9+25}}\\
 +
&&\\
 +
& = & \displaystyle{\sqrt{34}.}
 +
\end{array}</math>
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Using the formula in the Foundations section, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\hat{y}} & = & \displaystyle{\frac{\vec{y}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}\vec{v}}\\
 +
&&\\
 +
& = & \displaystyle{\frac{-1(2)+3(0)+0(5)}{(-1)(-1)+3(3)+0(0)}\begin{bmatrix}
 +
          -1 \\
 +
          3 \\
 +
          0
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|Continuing, we get
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\hat{y}} & = & \displaystyle{\frac{-2}{20}\begin{bmatrix}
 +
          -1 \\
 +
          3 \\
 +
          0
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          \frac{1}{10} \\
 +
          \frac{-3}{10} \\
 +
          0
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>\begin{bmatrix}
 +
          \frac{-1}{\sqrt{10}} \\
 +
          \frac{3}{\sqrt{10}} \\
 +
          0
 +
        \end{bmatrix}</math>
 +
|-
 +
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math>\sqrt{34}</math>
 
|-
 
|-
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math>\begin{bmatrix}
 +
          \frac{1}{10} \\
 +
          \frac{-3}{10} \\
 +
          0
 +
        \end{bmatrix}</math>
 
|}
 
|}
−
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+
[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 12:34, 15 October 2017

Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=\begin{bmatrix} -1 \\ 3 \\ 0 \end{bmatrix}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}=\begin{bmatrix} 2 \\ 0 \\ 5 \end{bmatrix}.}

(a) Find a unit vector in the direction of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}.}

(b) Find the distance between  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}.}

(c) Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\text{Span }\{\vec{v}\}.}   Compute the orthogonal projection of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}}   onto  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L.}

Foundations:  
1. The distance between the vectors  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dist}(\vec{u},\vec{v})=||\vec{u}-\vec{v}||.}
2. The orthogonal projection of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}}   onto  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{y}=\text{proj}_L \vec{y}=\frac{\vec{y}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}\vec{v}.}


Solution:

(a)

Step 1:  
First, we calculate  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||\vec{v}||.}  
We get

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{||\vec{v}||} & = & \displaystyle{\sqrt{(-1)^2+3^2+0^2}}\\ &&\\ & = & \displaystyle{\sqrt{1+9}}\\ &&\\ & = & \displaystyle{\sqrt{10}.} \end{array}}

Step 2:  
Now, to get a unit vector in the direction of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v},}   we take the vector  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}}   and divide by  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||\vec{v}||.}
Hence, we get the vector
       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\frac{1}{||\vec{v}||}\vec{v}} & = & \displaystyle{\frac{1}{\sqrt{10}}\begin{bmatrix} -1 \\ 3 \\ 0 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} \frac{-1}{\sqrt{10}} \\ \frac{3}{\sqrt{10}} \\ 0 \end{bmatrix}.} \end{array}}

(b)

Step 1:  
Using the formula in the Foundations section, we have

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\text{dist}(\vec{v},\vec{y})} & = & \displaystyle{||\vec{v}-\vec{y}||}\\ &&\\ & = & \displaystyle{\Bigg|\Bigg|\begin{bmatrix} -3 \\ 3 \\ -5 \end{bmatrix}\Bigg|\Bigg|.} \end{array}}

Step 2:  
Continuing, we get

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\text{dist}(\vec{v},\vec{y}) } & = & \displaystyle{\sqrt{(-3)^2+3^2+(-5)^2}}\\ &&\\ & = & \displaystyle{\sqrt{9+9+25}}\\ &&\\ & = & \displaystyle{\sqrt{34}.} \end{array}}

(c)

Step 1:  
Using the formula in the Foundations section, we have

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\hat{y}} & = & \displaystyle{\frac{\vec{y}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}\vec{v}}\\ &&\\ & = & \displaystyle{\frac{-1(2)+3(0)+0(5)}{(-1)(-1)+3(3)+0(0)}\begin{bmatrix} -1 \\ 3 \\ 0 \end{bmatrix}.} \end{array}}

Step 2:  
Continuing, we get

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\hat{y}} & = & \displaystyle{\frac{-2}{20}\begin{bmatrix} -1 \\ 3 \\ 0 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} \frac{1}{10} \\ \frac{-3}{10} \\ 0 \end{bmatrix}.} \end{array}}


Final Answer:  
   (a)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \frac{-1}{\sqrt{10}} \\ \frac{3}{\sqrt{10}} \\ 0 \end{bmatrix}}
   (b)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{34}}
   (b)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \frac{1}{10} \\ \frac{-3}{10} \\ 0 \end{bmatrix}}

Return to Review Problems

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