Chapter 2.2 Practice

Questions

Question 1

  1. Determine the coordinates of the following vectors.
    1. \(4V\) where \(V=\langle -1,2\rangle\)
    2. \(-V\) where \(V=\langle 4,6\rangle\)
    3. \(2V\) where \(V=\langle -3,5\rangle\)
    4. \(-7V\) where \(V=\langle -2,-1\rangle\)

Question 2

  1. Calculate the length of the following vectors.
    1. \(V=\langle -1,2\rangle\) and \(4V\)
    2. \(V=\langle 4,6\rangle\) and \(-V\)
    3. \(V=\langle -3,5\rangle\) and \(2V\)
    4. \(V=\langle -2,-1\rangle\) and \(-7V\)

Question 3

  1. Write the polar form of the following vectors.
    1. \(V=\langle -1,2\rangle\)
    2. \(V=\langle 4,6\rangle\)
    3. \(V=\langle -3,5\rangle\)
    4. \(V=\langle -2,-1\rangle\)

Question 4

  1. Determine an equation for the circle with following radius and center.
    1. \((5,1)\) and \(r=2\).
    2. \((-3,2)\) and \(r=9\).
    3. \((2,4)\) and \(r=3\).
    4. \((-1,-8)\) and \(r=\sqrt{5}\).
    5. \((3,0)\) and \(r=1\).

Question 5

  1. Identify the center and radius of the following circles whose equation is given below:
    1. \(x^2 - 2 x + y^2 + 8 y + 13 = 0\)
    2. \(x^2 + 6 x + y^2 + 10 y + 34 = 2\)
    3. \(x^2 + y^2 - 2 y - 15 = 0\)
    4. \(x^2 + 14 x + y^2 + 24 = 0\)

Question 6

  1. Determine all points at which the line given by the following equation intersects the unit circle:
    1. \(y=8x\)
    2. \(y=-3x\)
    3. \(y=4x\)
    4. \(y=-5x\)

Question 7

  1. Take \(C\) to be the circle with radius \(2\) and center \((-1,2).\) Take \(L\) to be the line that intersects \((-1,2)\) and \((3,4).\) Determine all points at which \(L\) intersects \(C\).

Question 8

  1. Identify the projection of the given point \(p\) onto the unit circle.
    1. \((4,6)\)
    2. \((-1,-3)\)
    3. \((-5,-2)\)
    4. \((0,8)\)

Question 9

  1. For each function \(f\), write its transformed function \(g\) as a composite function using the functions \(T_h\) and \(S_a\) given by \[T_h(x)=x+h\quad\text{and}\quad S_a(x)=ax.\]
    1. \(f(x)=x^2\), \(g(x)=3(x+2)^2+1\)
    2. \(f(x)=x^3\), \(g(x)=-(3x-2)^3-2\)
    3. \(f(x)=|x|\), \(g(x)=2|5x|+1\)
    4. \(f(x)=\frac{1}{x}\), \(g(x)=\frac{2}{5x-1}+3.\)

Answers

Question 1

    1. \(\langle -4,8\rangle\)
    2. \(\langle -4,-6\rangle\)
    3. \(\langle -6,10\rangle\)
    4. \(\langle 14,7\rangle\)

Question 2

    1. \(\|V\|=\sqrt{5}\) and \(\|4V\|=\sqrt{80}\)
    2. \(\|V\|=\sqrt{52}\) and \(\|-V\|=\sqrt{52}\)
    3. \(\|V\|=\sqrt{34}\) and \(\|2V\|=\sqrt{136}\)
    4. \(\|V\|=\sqrt{5}\) and \(\|-7V\|=\sqrt{245}\)

Question 3

    1. \(\sqrt{5}\left\langle \frac{-1}{\sqrt{5}},\frac{2}{\sqrt{5}}\right\rangle\)
    2. \(\sqrt{52}\left\langle \frac{4}{\sqrt{52}},\frac{6}{\sqrt{52}}\right\rangle\)
    3. \(\sqrt{34}\left\langle \frac{-3}{\sqrt{34}},\frac{5}{\sqrt{34}}\right\rangle\)
    4. \(\sqrt{5}\left\langle \frac{-2}{\sqrt{5}},\frac{-1}{\sqrt{5}}\right\rangle\)

Question 4

    1. \((x-5)^2+(y-1)^2=4\)
    2. \((x+3)^2+(y-2)^2=81\)
    3. \((x-2)^2+(y-4)^2=9\)
    4. \((x+1)^2+(y+8)^2=5\)
    5. \((x-3)^2+y^2=1\)

Question 5

    1. center at \((1,-4)\), radius \(2\)
    2. center at \((-3,-5)\), radius \(\sqrt{2}\)
    3. center at \((0,1)\), radius \(4\)
    4. center at \((-7,0)\), radius \(5\)

Question 6

    1. \(\left(\frac{1}{\sqrt{65}},\frac{8}{\sqrt{65}}\right)\), \(\left(-\frac{1}{\sqrt{65}},-\frac{8}{\sqrt{65}}\right)\)
    2. \(\left(\frac{1}{\sqrt{10}},-\frac{3}{\sqrt{10}}\right)\), \(\left(-\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right)\)
    3. \(\left(\frac{1}{\sqrt{17}},\frac{4}{\sqrt{17}}\right)\), \(\left(-\frac{1}{\sqrt{17}},-\frac{4}{\sqrt{17}}\right)\)
    4. \(\left(\frac{1}{\sqrt{26}},-\frac{5}{\sqrt{26}}\right)\), \(\left(-\frac{1}{\sqrt{26}},\frac{5}{\sqrt{26}}\right)\)

Question 7

  1. \(\left(\frac{4}{\sqrt{5}}-1,\frac{2}{\sqrt{5}}+2\right)\), \(\left(-\frac{4}{\sqrt{5}}-1,-\frac{2}{\sqrt{5}}+2\right)\)

Question 8

    1. \(\left(\frac{4}{\sqrt{52}},\frac{6}{\sqrt{52}}\right)\), \(\left(-\frac{4}{\sqrt{52}},-\frac{6}{\sqrt{52}}\right)\)
    2. \(\left(-\frac{1}{\sqrt{10}},-\frac{3}{\sqrt{10}}\right)\), \(\left(\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right)\)
    3. \(\left(-\frac{5}{\sqrt{29}},-\frac{2}{\sqrt{29}}\right)\), \(\left(\frac{5}{\sqrt{29}},\frac{2}{\sqrt{29}}\right)\)
    4. \(\left(0,1\right)\), \(\left(0,-1\right)\)

Question 9

    1. \(T_1\circ S_3\circ f\circ T_2\)
    2. \(T_{-2}\circ S_{-1} \circ f \circ S_3 \circ T_{-\frac{2}{3}}\)
    3. \(T_1\circ S_2\circ f\circ S_5\)
    4. \(T_3\circ S_2\circ f\circ S_5\circ T_{-1}\)

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