22.3: Exercises
- Page ID
- 54474
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Graph the vectors in the plane.
- \(\overrightarrow{PQ}\) with \(P(2,1)\) and \(Q(4,7)\)
- \(\overrightarrow{PQ}\) with \(P(-3,3)\) and \(Q(-5,-4)\)
- \(\overrightarrow{PQ}\) with \(P(0,-4)\) and \(Q(6,0)\)
- \(\langle -2,4 \rangle\)
- \(\langle -3,-3 \rangle\)
- \(\langle 5,5\sqrt{2} \rangle\)
- Answer
-
Find the magnitude and directional angle of the vector.
- \(\langle 6,8\rangle\)
- \(\langle -2,5 \rangle\)
- \(\langle -4,-4\rangle\)
- \(\langle 3,-3\rangle\)
- \(\langle 2,-2 \rangle\)
- \(\langle 4\sqrt{3},4\rangle\)
- \(\langle -\sqrt{3},-1\rangle\)
- \(\langle -4,4\sqrt{3}\rangle\)
- \(\langle -2\sqrt{3},-2\rangle\)
- \(\overrightarrow{PQ}\), where \(P(3,1)\) and \(Q(7,4)\)
- \(\overrightarrow{PQ}\), where \(P(4,-2)\) and \(Q(-5,7)\)
- Answer
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- \(10,53^{\circ}\)
- \(\sqrt{29}, 112^{\circ}\)
- \(4 \sqrt{2}, 225^{\circ}\)
- \(3 \sqrt{2},-45^{\circ}\)
- \(2 \sqrt{2},-45^{\circ}\)
- \(8,30^{\circ}\)
- \(2,210^{\circ}\)
- \(8,120^{\circ}\)
- \(4,210^{\circ}\)
- \(5,37^{\circ}\)
- \(9 \sqrt{2}, 135^{\circ}\)
Perform the operation on the vectors.
- \(5\cdot \langle 3,2\rangle\)
- \(2\cdot \langle -1,4 \rangle\)
- \((-10)\cdot \langle -\dfrac{3}{2},-\dfrac{7}{5} \rangle\)
- \(\langle2,3\rangle+\langle 6,1\rangle\)
- \(\langle 5,-4\rangle-\langle -8,-9\rangle\)
- \(3\cdot \langle 5,3\rangle+4\cdot \langle 2,8\rangle\)
- \((-2) \langle -5,-4\rangle-6\langle -1,-2\rangle\)
- \(\dfrac 2 3 \langle-3,6\rangle-\dfrac 7 5\langle10,-15\rangle\)
- \(\sqrt{2}\cdot \langle \dfrac{\sqrt{8}}{6},\dfrac{-5\sqrt{2}}{12} \rangle-2\langle\dfrac 2 3,\dfrac 5 3\rangle\)
- \(6\vec{i}-4\vec{j}\)
- \(-5\vec{i}+\vec{j}+3\vec{i}\)
- \(3\cdot\langle-4,2\rangle-8\vec{j}+12\vec{i}\)
- Find \(4\vec{v}+7\vec{w}\) for \(\vec{v}=\langle 2,3 \rangle\) and \(\vec{w}=\langle 5,1\sqrt{3} \rangle\)
- Find \(\vec{v}-2\vec{w}\) for \(\vec{v}=\langle -11,-6 \rangle\) and \(\vec{w}=\langle -3,2 \rangle\)
- Find \(3\vec{v}-\vec{w}\) for \(\vec{v}=-4\vec{i}+7\vec{j}\) and \(\vec{w}=6\vec{i}+\vec{j}\)
- Find \(-\vec{v}-\sqrt{5}\vec{w}\) for \(\vec{v}=5\vec{j}\) and \(\vec{w}=-8\vec{i}+\sqrt{5}\vec{j}\)
- Answer
-
- \(\langle 15,10\rangle\)
- \(\langle-2,8\rangle\)
- \(\langle 15,14\rangle\)
- \(\langle 8,4\rangle\)
- \(\langle 13,5\rangle\)
- \(\langle 23,41\rangle\)
- \(\langle 16,20\rangle\)
- \(\langle-16,25\rangle\)
- \(\left\langle-\dfrac{2}{3},-\dfrac{25}{6}\right\rangle\)
- \(\langle 6,-4\rangle\)
- \(\langle-2,1\rangle\)
- \(\langle 0,-2\rangle\)
- \(\langle 43,12+7 \sqrt{3}\rangle\)
- \(\langle-5,-10\rangle\)
- \(\langle-18,20\rangle\)
- \(\langle 8 \sqrt{5},-10\rangle\)
Find a unit vector in the direction of the given vector.
- \(\langle 8,-6 \rangle\)
- \(\langle -3,-\sqrt{7} \rangle\)
- \(\langle 9,2 \rangle\)
- \(\langle -\sqrt{5},\sqrt{31} \rangle\)
- \(\langle 5\sqrt{2},3\sqrt{10} \rangle\)
- \(\langle 0,-\dfrac{3}{5} \rangle\)
- Answer
-
- \(\left\langle\dfrac{4}{5},-\dfrac{3}{5}\right\rangle\)
- \(\left\langle-\dfrac{3}{4},-\dfrac{\sqrt{7}}{4}\right\rangle\)
- \(\left\langle\dfrac{9 \sqrt{85}}{85}, \dfrac{2 \sqrt{85}}{85}\right\rangle\)
- \(\left\langle-\dfrac{\sqrt{5}}{6}, \dfrac{\sqrt{31}}{6}\right\rangle\)
- \(\left\langle\dfrac{5 \sqrt{70}}{70}, \dfrac{3 \sqrt{14}}{14}\right\rangle\)
- \(\langle 0,-1\rangle\)
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).
- \(||\vec{v_1}||=6\), and \(\theta_1=60^\circ\), and \(||\vec{v_2}||=2\), and \(\theta_2=180^\circ\)
- \(||\vec{v_1}||=3.7\), and \(\theta_1=92^\circ\), and \(||\vec{v_2}||=2.2\), and \(\theta_2=253^\circ\)
- \(||\vec{v_1}||=8\), and \(\theta_1=\dfrac{3\pi}{4}\), and \(||\vec{v_2}||=8\sqrt{2}\), and \(\theta_2=\dfrac{3\pi}{2}\)
- Answer
-
- \(\vec{v}=\langle 1,3 \sqrt{3}\rangle,\|\vec{v}\|=2 \sqrt{7}, \theta \approx 79^{\circ}\)
- \(\vec{v} \approx\langle-.772,1.594\rangle,\|\vec{v}\| \approx 7.63, \theta \approx 116^{\circ}\)
- \(\vec{v}=\langle-4 \sqrt{2},-4 \sqrt{2}\rangle,\|v\|=8, \theta=225^{\circ}=\dfrac{5 \pi}{4}\)