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

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1:    
 
!Step 1:    
 +
|-
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|First, we calculate &nbsp;<math>||\vec{v}||.</math>&nbsp;
 +
|-
 +
|We get
 
|-
 
|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{||\vec{v}||} & = & \displaystyle{\sqrt{(-1)^2+3^2+0^2}}\\
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&&\\
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& = & \displaystyle{\sqrt{1+9}}\\
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&&\\
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& = & \displaystyle{\sqrt{10}.}
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\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>\vec{v},</math>&nbsp; we take the vector &nbsp;<math>\vec{v}</math>&nbsp; and divide by &nbsp;<math>||\vec{v}||.</math>
 +
|-
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|Hence, we get the vector
 +
|-
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|&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\frac{1}{||\vec{v}||}\vec{v}} & = & \displaystyle{\frac{1}{\sqrt{10}}\begin{bmatrix}
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          -1 \\
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          3 \\
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          0
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        \end{bmatrix}}\\
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&&\\
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& = & \displaystyle{\begin{bmatrix}
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          \frac{-1}{\sqrt{10}} \\
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          \frac{3}{\sqrt{10}} \\
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          0
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        \end{bmatrix}.}
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\end{array}</math>
 
|}
 
|}
  

Revision as of 14:04, 12 October 2017

Let  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.}

(a) Find a unit vector in the direction of  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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{u}}{\vec{u}\cdot \vec{u}}\vec{u}.}


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:  
Step 2:  

(c)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

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