Chapter 2.3 Practice

Questions

Question 1

  1. For each line \(L\), identify a vector \(V\) that moves points along \(L\)
    1. the line \(L\) with slope \(8\) and \(y\)-intercept
    2. the line \(L\) that passes through the points \((2,5)\) and \((1,-2)\)
    3. the line \(L\) whose equation is given by \(y=-\frac{3}{4}x+1\)

Question 2

  1. For each line \(L\), find all points in \(L\) that are a distance of \(k\) from the given point \(p\).
    1. the line \(L\) that passes through the points \((1,3)\) and \((5,4)\); all points that are distance of \(3\) from the point \((1,3)\)
    2. the line \(L\) that passes through the points \((2,7)\) and \((0,1)\); all points that are distance of \(7\) from the point \((0,1)\)
    3. the line \(L\) given by the equation \(y=3x+1\); all points that are distance of \(2\) from the point \((1,4)\)

Question 3

  1. For each line or line segment \(L\), find the point on \(L\) that is a distance of \(k\) from the point or points indicated.
    1. the line segment \(L\) with end points \((-1,-2)\) and \((1,4)\); all points \(p\) in \(L\) so that the distance between \(p\) to \((-1,-2)\) is \(\frac{1}{3}\) times the distance between \((-1,-2)\) and \((1,4)\)
    2. the line \(L\) that passes through \((1,10)\) and \((4,3)\); all points \(p\) in \(L\) so that the distance between \(p\) to \((4,3)\) is \(2\) times the distance between \((1,10)\) and \((4,3)\)
    3. the line segment \(L\) with end points \((-1,3)\) and \((5,8)\); all points \(p\) in \(L\) so that the distance between \(p\) to \((-1,3)\) is \(5\) times the distance between \(p\) and \((5,8)\)
    4. the line segment \(L\) with end points \((-3,-2)\) and \((-1,-7)\); all points \(p\) in \(L\) so that the distance between \(p\) to \((-1,-7)\) is \(\frac{1}{10}\) times the distance between \(p\) and \((-3,-2)\)

Question 4

  1. For each function \(\ell\) that describes the position of a particle, write the equation for the line along which the particle moves in point-slope form.
    1. \(\ell(t)=t\langle 2,-3\rangle +(1,3)\)
    2. \(\ell(t)=t\langle -1,5\rangle +(2,5)\)
    3. \(\ell(t)=t\langle 6,4\rangle +(-1,-6)\)

Question 5

  1. Determine an equation for the position of a particle with the given characteristics.
    1. the particle moves with a constant velocity with a speed of 4, is at \((2,5)\) at \(t=1\) and intersects \((-2,-4)\)
    2. the particle moves with a constant velocity, is at \((-2,1)\) at \(t=3\) and is at \((5,8)\) at \(t=8\)
    3. the particle moves with constant velocity on \([0,3]\) and \((3,6]\), is at \((1,2)\) at time \(t=0\), is at \((-2,5)\) at time \(t=3\) and is at \((4,6)\) at time \(t=6\)
    4. the particle moves with constant velocity on \([1,5]\) and \((5,10]\), is at \((2,1)\) at time \(t=1\), is at \((5,6)\) at time \(t=5\), intersects the point \((7,1)\) for some time \(t>5\) and has speed 2 on the time interval \((5,10].\)

Answers

Question 1

    1. \(V=\langle 1,8\rangle\)
    2. \(V=\langle -1,-7 \rangle\)
    3. \(V=\langle 4,-3 \rangle\) or \(V=\langle 1,-\frac{3}{4}\rangle\)

Question 2

    1. \(p=\left\langle \frac{12}{\sqrt{17}}+1,\frac{3}{\sqrt{17}}+3\right\rangle\) and \(q=\left\langle -\frac{12}{\sqrt{17}}+1,-\frac{3}{\sqrt{17}}+3\right\rangle\)
    2. \(p=\left(-\frac{14}{\sqrt{40}},-\frac{42}{\sqrt{40}}+1\right)\) and \(q=\left(\frac{14}{\sqrt{40}},\frac{42}{\sqrt{40}}+1\right)\)
    3. \(p=\left(\frac{2}{\sqrt{10}}+1,\frac{6}{\sqrt{10}}+4\right)\) and \(q=\left(-\frac{2}{\sqrt{10}}+1,-\frac{6}{\sqrt{10}}+4\right)\)

Question 3

    1. \(\left(-\frac{1}{3},0\right)\)
    2. \(\left(10,-11\right)\)
    3. \(\left(4,\frac{43}{6}\right)\)
    4. \(\left(-\frac{13}{11},-\frac{72}{11}\right)\)

Question 4

    1. \(y=-\frac{3}{2}(x-1)+3\)
    2. \(y=-5(x-2)+5\)
    3. \(y=\frac{2}{3}(x+1)-6\)

Question 5

    1. \(\ell(t)=\frac{4(t-1)}{\sqrt{97}}\langle -4,-9\rangle+(2,5)\)
    2. \(\ell(t)=\frac{t-3}{5}\langle 7,7\rangle+(-2,1)\)
    3. \(\ell(t)=\begin{cases}\frac{t}{3}\langle -3 ,3\rangle +(1,2)&\text{if }0\leq t\leq 3\\ \frac{t-3}{3}\langle 6 ,1\rangle +(-2,5)&\text{if } 3<t\leq 6\end{cases}\)
    4. \(\ell(t)=\begin{cases}\frac{t-1}{4}\langle 3 ,5\rangle +(2,1)&\text{if }1\leq t\leq 5\\ 2(t-5)\left\langle \frac{2}{\sqrt{29}} ,\frac{-5}{\sqrt{29}}\right\rangle +(5,6)&\text{if } 5<t\leq 10\end{cases}\)

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