10.8E: Vectors (Exercises)
- Page ID
- 56122
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, determine whether the two vectors, \(\mathbf{u}\) and \(\mathbf{v},\) are equal, where \(\mathbf{u}\) has an initial point \(P_{1}\) and a terminal point \(P_{2},\) and \(\mathbf{v}\) has an initial point \(P_{3}\) and a terminal point \(P_{4}\).
52. \(P_{1}=(-1,4), P_{2}=(3,1), P_{3}=(5,5)\) and \(P_{4}=(9,2)\)
53. \(P_{1}=(6,11), P_{2}=(-2,8), P_{3}=(0,-1)\) and \(P_{4}=(-8,2)\)
For the following exercises, use the vectors \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \mathbf{v}=4 \mathbf{i}-3 \mathbf{j},\) and \(w=-2 \mathbf{i}+5 \mathbf{j}\) to evaluate the expression.
54. \(u-v\)
55. \(2 v-u+w\)
For the following exercises, find a unit vector in the same direction as the given vector.
56. \(a=8 i-6 j\)
57. \(b=-3 i-j\)
For the following exercises, find the magnitude and direction of the vector.
58. \(\langle 6,-2\rangle\)
59. \(\langle-3,-3\rangle\)
For the following exercises, calculate \(\mathbf{u} \cdot \mathbf{v}\).
60. \(u=-2 i+j\) and \(v=3 i+7 j\)
61. \(u=i+4 j\) and \(v=4 i+3 j\)
62. Given \(\boldsymbol{v}=\langle-3,4\rangle\) draw \(\boldsymbol{v}, 2 \boldsymbol{v},\) and \(\frac{1}{2} \boldsymbol{v}\)
63. Given the vectors shown in Figure 4 , sketch \(\boldsymbol{u}+\boldsymbol{v}, \boldsymbol{u}-\boldsymbol{v}\) and \(3 \boldsymbol{v}\).
Figure 4
64. Given initial point \(P_{1}=(3,2)\) and terminal point \(P_{2}=(-5,-1),\) write the vector \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). Draw the points and the vector on the graph.