Chapter 2.4 Practice

Questions

Question 1

  1. For each vector \(V,\) write out \(V_{\perp}.\)
    1. \(V=\langle 1,6\rangle\)
    2. \(V=\langle -2,8\rangle\)
    3. \(V=\langle 9,-4\rangle\)
    4. \(V=\langle -10,-15\rangle\)

Question 2

  1. For each line \(L\), identify the line \(L_{\perp}\) that is perpendicular to \(L\) and intersects at the indicated point.
    1. the line \(L\) given by the equation \(y=\frac{2}{5}x+2\); the perpendicular line must pass through the point \((1,2)\)
    2. the line \(L\) that passes through the points \((0,5)\) and \((1,2)\); the perpendicular line must pass through the point \((3,8)\)
    3. the line \(L\) that passes through the points \((-3,-1)\) and \((2,-4)\); the perpendicular line must pass through the \(y\) intercept of \(L\)
    4. the line \(L\) given by \(L=\{t\langle 2,6\rangle+(1,2)\colon t\in\mathbb{R}\}\); the perpendicular line must pass through the \(x\) intercept of \(L\)

Question 3

  1. For each line \(L\), find the point on \(L\) that is closest to the indicated point.
    1. the line \(L\) that passes through \((-7,5)\) and \((-3,6)\); find the point on \(L\) closest to \((-2,5)\)
    2. the line \(L\) given by \(y=8x+1\); find the point on \(L\) closest to \((1,6)\)
    3. the line \(L\) given by \(L=\{t\langle -1,5\rangle+(3,-1)\colon t\in\mathbb{R}\}\); find the point on \(L\) closest to \((2,0)\)

Question 4

  1. For each line \(L\) and point \(p\), find the reflection of \(p\) across \(L.\)
    1. the line \(L\) that passes through \((-7,5)\) and \((-3,6)\); reflect \(p=(-2,5)\) across \(L\)
    2. the line \(L\) given by \(y=8x+1\); reflect \(p=(1,6)\) across \(L\)
    3. the line \(L\) given by \(L=\{t\langle -1,5\rangle+(3,-1)\colon t\in\mathbb{R}\}\); reflect \(p=(2,0)\) across \(L\)

Answers

Question 1

    1. \(V_{\perp}=\langle -6,1\rangle\)
    2. \(V_{\perp}=\langle -8,-2\rangle\)
    3. \(V_{\perp}=\langle 4,9\rangle\)
    4. \(V_{\perp}=\langle 15,-10\rangle\)

Question 2

    1. \(L_\perp\) is given by the equation \(y=-\frac{5}{2}(x-1)+2\)
    2. \(L_\perp\) is given by the equation \(y=\frac{1}{3}(x-3)+8\)
    3. \(L_\perp\) is given by the equation \(y=\frac{5}{3}x-\frac{14}{5}\)
    4. \(L_\perp\) is given by the equation \(y=-\frac{1}{3}\left(x-\frac{1}{3}\right)\)

Question 3

    1. \(\left(-\frac{39}{17},\frac{105}{17}\right)\)
    2. \(\left(\frac{41}{65},\frac{393}{65}\right)\)
    3. \(\left(\frac{36}{13},\frac{2}{13}\right)\)

Question 4

    1. \(\left(-\frac{44}{17},\frac{125}{17}\right)\)
    2. \(\left(\frac{17}{65},\frac{396}{65}\right)\)
    3. \(\left(\frac{46}{13},\frac{4}{13}\right)\)

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