Difference between revisions of "031 Review Part 2, Problem 6"
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
||
| Line 14: | Line 14: | ||
<span class="exam">(c) Let <math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math> Compute the orthogonal projection of <math style="vertical-align: -3px">\vec{y}</math> onto <math style="vertical-align: 0px">L.</math> | <span class="exam">(c) Let <math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math> Compute the orthogonal projection of <math style="vertical-align: -3px">\vec{y}</math> onto <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;" | ||
| Line 27: | Line 26: | ||
|- | |- | ||
| | | | ||
| − | ::<math>\hat{y}=\text{proj}_L \vec{y}=\frac{\vec{y}\cdot \vec{ | + | ::<math>\hat{y}=\text{proj}_L \vec{y}=\frac{\vec{y}\cdot \vec{v}}{\vec{v}\cdot \vec{v}}\vec{v}.</math> |
|} | |} | ||
| Line 141: | Line 140: | ||
&&\\ | &&\\ | ||
& = & \displaystyle{\begin{bmatrix} | & = & \displaystyle{\begin{bmatrix} | ||
| − | \frac{1}{ | + | \frac{1}{10} \\ |
| − | + | \frac{-3}{10} \\ | |
0 | 0 | ||
\end{bmatrix}.} | \end{bmatrix}.} | ||
| Line 161: | Line 160: | ||
|- | |- | ||
| '''(b)''' <math>\begin{bmatrix} | | '''(b)''' <math>\begin{bmatrix} | ||
| − | \frac{1}{ | + | \frac{1}{10} \\ |
| − | + | \frac{-3}{10} \\ | |
0 | 0 | ||
\end{bmatrix}</math> | \end{bmatrix}</math> | ||
|} | |} | ||
| − | [[031_Review_Part_2|'''<u>Return to | + | [[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 |
|
| 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 |
|
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}} |