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22.3: Ejercicios

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    Ejercicio\(\PageIndex{1}\)

    Grafica los vectores en el plano.

    1. \(\overrightarrow{PQ}\)con\(P(2,1)\) y\(Q(4,7)\)
    2. \(\overrightarrow{PQ}\)con\(P(-3,3)\) y\(Q(-5,-4)\)
    3. \(\overrightarrow{PQ}\)con\(P(0,-4)\) y\(Q(6,0)\)
    4. \(\langle -2,4 \rangle\)
    5. \(\langle -3,-3 \rangle\)
    6. \(\langle 5,5\sqrt{2} \rangle\)
    Contestar

    clipboard_e2d954a3c45d87071c44e6249735cfa7d.png

    Ejercicio\(\PageIndex{2}\)

    Encuentra la magnitud y el ángulo direccional del 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. \(\overrightarrow{PQ}\), dónde\(P(3,1)\) y\(Q(7,4)\)
    11. \(\overrightarrow{PQ}\), dónde\(P(4,-2)\) y\(Q(-5,7)\)
    Contestar
    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. \(9 \sqrt{2}, 135^{\circ}\)

    Ejercicio\(\PageIndex{3}\)

    Realizar la operación en los vectores.

    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. Encuentra\(4\vec{v}+7\vec{w}\) para\(\vec{v}=\langle 2,3 \rangle\) y\(\vec{w}=\langle 5,1\sqrt{3} \rangle\)
    14. Encuentra\(\vec{v}-2\vec{w}\) para\(\vec{v}=\langle -11,-6 \rangle\) y\(\vec{w}=\langle -3,2 \rangle\)
    15. Encuentra\(3\vec{v}-\vec{w}\) para\(\vec{v}=-4\vec{i}+7\vec{j}\) y\(\vec{w}=6\vec{i}+\vec{j}\)
    16. Encuentra\(-\vec{v}-\sqrt{5}\vec{w}\) para\(\vec{v}=5\vec{j}\) y\(\vec{w}=-8\vec{i}+\sqrt{5}\vec{j}\)
    Contestar
    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. \(\langle 8 \sqrt{5},-10\rangle\)

    Ejercicio\(\PageIndex{4}\)

    Encuentra un vector unitario en la dirección del vector dado.

    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. \(\langle 0,-\dfrac{3}{5} \rangle\)
    Contestar
    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. \(\langle 0,-1\rangle\)

    Ejercicio\(\PageIndex{5}\)

    Se están agregando los vectores\(\vec{v_1}\) y a\(\vec{v_2}\) continuación. Encuentra la magnitud aproximada y el ángulo direccional de la suma\(\vec{v}=\vec{v_1}+\vec{v_2}\) (ver Ejemplo 22.2.4).

    1. \(||\vec{v_1}||=6\), y\(\theta_1=60^\circ\), y\(||\vec{v_2}||=2\), y\(\theta_2=180^\circ\)
    2. \(||\vec{v_1}||=3.7\), y\(\theta_1=92^\circ\), y\(||\vec{v_2}||=2.2\), y\(\theta_2=253^\circ\)
    3. \(||\vec{v_1}||=8\), y\(\theta_1=\dfrac{3\pi}{4}\), y\(||\vec{v_2}||=8\sqrt{2}\), y\(\theta_2=\dfrac{3\pi}{2}\)
    Contestar
    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: Ejercicios 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.