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22.3: Exercises

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    Exercise \(\PageIndex{1}\)

    Graph the vectors in the plane.

    1. \(\overrightarrow{PQ}\) with \(P(2,1)\) and \(Q(4,7)\)
    2. \(\overrightarrow{PQ}\) with \(P(-3,3)\) and \(Q(-5,-4)\)
    3. \(\overrightarrow{PQ}\) with \(P(0,-4)\) and \(Q(6,0)\)
    4. \(\langle -2,4 \rangle\)
    5. \(\langle -3,-3 \rangle\)
    6. \(\langle 5,5\sqrt{2} \rangle\)
    Answer

    clipboard_e2d954a3c45d87071c44e6249735cfa7d.png

    Exercise \(\PageIndex{2}\)

    Find the magnitude and directional angle of the vector.

    1. \(\langle 6,8\rangle\)
    2. \(\langle -2,5 \rangle\)
    3. \(\langle -4,-4\rangle\)
    4. \(\langle 3,-3\rangle\)
    5. \(\langle 2,-2 \rangle\)
    6. \(\langle 4\sqrt{3},4\rangle\)
    7. \(\langle -\sqrt{3},-1\rangle\)
    8. \(\langle -4,4\sqrt{3}\rangle\)
    9. \(\langle -2\sqrt{3},-2\rangle\)
    10. \(\overrightarrow{PQ}\), where \(P(3,1)\) and \(Q(7,4)\)
    11. \(\overrightarrow{PQ}\), where \(P(4,-2)\) and \(Q(-5,7)\)
    Answer
    1. \(10,53^{\circ}\)
    2. \(\sqrt{29}, 112^{\circ}\)
    3. \(4 \sqrt{2}, 225^{\circ}\)
    4. \(3 \sqrt{2},-45^{\circ}\)
    5. \(2 \sqrt{2},-45^{\circ}\)
    6. \(8,30^{\circ}\)
    7. \(2,210^{\circ}\)
    8. \(8,120^{\circ}\)
    9. \(4,210^{\circ}\)
    10. \(5,37^{\circ}\)
    11. \(9 \sqrt{2}, 135^{\circ}\)

    Exercise \(\PageIndex{3}\)

    Perform the operation on the vectors.

    1. \(5\cdot \langle 3,2\rangle\)
    2. \(2\cdot \langle -1,4 \rangle\)
    3. \((-10)\cdot \langle -\dfrac{3}{2},-\dfrac{7}{5} \rangle\)
    4. \(\langle2,3\rangle+\langle 6,1\rangle\)
    5. \(\langle 5,-4\rangle-\langle -8,-9\rangle\)
    6. \(3\cdot \langle 5,3\rangle+4\cdot \langle 2,8\rangle\)
    7. \((-2) \langle -5,-4\rangle-6\langle -1,-2\rangle\)
    8. \(\dfrac 2 3 \langle-3,6\rangle-\dfrac 7 5\langle10,-15\rangle\)
    9. \(\sqrt{2}\cdot \langle \dfrac{\sqrt{8}}{6},\dfrac{-5\sqrt{2}}{12} \rangle-2\langle\dfrac 2 3,\dfrac 5 3\rangle\)
    10. \(6\vec{i}-4\vec{j}\)
    11. \(-5\vec{i}+\vec{j}+3\vec{i}\)
    12. \(3\cdot\langle-4,2\rangle-8\vec{j}+12\vec{i}\)
    13. Find \(4\vec{v}+7\vec{w}\) for \(\vec{v}=\langle 2,3 \rangle\) and \(\vec{w}=\langle 5,1\sqrt{3} \rangle\)
    14. Find \(\vec{v}-2\vec{w}\) for \(\vec{v}=\langle -11,-6 \rangle\) and \(\vec{w}=\langle -3,2 \rangle\)
    15. Find \(3\vec{v}-\vec{w}\) for \(\vec{v}=-4\vec{i}+7\vec{j}\) and \(\vec{w}=6\vec{i}+\vec{j}\)
    16. Find \(-\vec{v}-\sqrt{5}\vec{w}\) for \(\vec{v}=5\vec{j}\) and \(\vec{w}=-8\vec{i}+\sqrt{5}\vec{j}\)
    Answer
    1. \(\langle 15,10\rangle\)
    2. \(\langle-2,8\rangle\)
    3. \(\langle 15,14\rangle\)
    4. \(\langle 8,4\rangle\)
    5. \(\langle 13,5\rangle\)
    6. \(\langle 23,41\rangle\)
    7. \(\langle 16,20\rangle\)
    8. \(\langle-16,25\rangle\)
    9. \(\left\langle-\dfrac{2}{3},-\dfrac{25}{6}\right\rangle\)
    10. \(\langle 6,-4\rangle\)
    11. \(\langle-2,1\rangle\)
    12. \(\langle 0,-2\rangle\)
    13. \(\langle 43,12+7 \sqrt{3}\rangle\)
    14. \(\langle-5,-10\rangle\)
    15. \(\langle-18,20\rangle\)
    16. \(\langle 8 \sqrt{5},-10\rangle\)

    Exercise \(\PageIndex{4}\)

    Find a unit vector in the direction of the given vector.

    1. \(\langle 8,-6 \rangle\)
    2. \(\langle -3,-\sqrt{7} \rangle\)
    3. \(\langle 9,2 \rangle\)
    4. \(\langle -\sqrt{5},\sqrt{31} \rangle\)
    5. \(\langle 5\sqrt{2},3\sqrt{10} \rangle\)
    6. \(\langle 0,-\dfrac{3}{5} \rangle\)
    Answer
    1. \(\left\langle\dfrac{4}{5},-\dfrac{3}{5}\right\rangle\)
    2. \(\left\langle-\dfrac{3}{4},-\dfrac{\sqrt{7}}{4}\right\rangle\)
    3. \(\left\langle\dfrac{9 \sqrt{85}}{85}, \dfrac{2 \sqrt{85}}{85}\right\rangle\)
    4. \(\left\langle-\dfrac{\sqrt{5}}{6}, \dfrac{\sqrt{31}}{6}\right\rangle\)
    5. \(\left\langle\dfrac{5 \sqrt{70}}{70}, \dfrac{3 \sqrt{14}}{14}\right\rangle\)
    6. \(\langle 0,-1\rangle\)

    Exercise \(\PageIndex{5}\)

    The vectors \(\vec{v_1}\) and \(\vec{v_2}\) below are being added. Find the approximate magnitude and directional angle of sum \(\vec{v}=\vec{v_1}+\vec{v_2}\) (see Example 22.2.4).

    1. \(||\vec{v_1}||=6\), and \(\theta_1=60^\circ\), and \(||\vec{v_2}||=2\), and \(\theta_2=180^\circ\)
    2. \(||\vec{v_1}||=3.7\), and \(\theta_1=92^\circ\), and \(||\vec{v_2}||=2.2\), and \(\theta_2=253^\circ\)
    3. \(||\vec{v_1}||=8\), and \(\theta_1=\dfrac{3\pi}{4}\), and \(||\vec{v_2}||=8\sqrt{2}\), and \(\theta_2=\dfrac{3\pi}{2}\)
    Answer
    1. \(\vec{v}=\langle 1,3 \sqrt{3}\rangle,\|\vec{v}\|=2 \sqrt{7}, \theta \approx 79^{\circ}\)
    2. \(\vec{v} \approx\langle-.772,1.594\rangle,\|\vec{v}\| \approx 7.63, \theta \approx 116^{\circ}\)
    3. \(\vec{v}=\langle-4 \sqrt{2},-4 \sqrt{2}\rangle,\|v\|=8, \theta=225^{\circ}=\dfrac{5 \pi}{4}\)

    This page titled 22.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.