1.3.E: The Cross Product (Exercises)
- Page ID
- 77664
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
For each of the following pairs of vectors \(\mathbf{x}\) and \(\mathbf{y}\), find \(\mathbf{x} \times \mathbf{y}\) and verify that \(\mathbf{x} \perp(\mathbf{x} \times \mathbf{y})\) and \(\mathbf{y} \perp(\mathbf{x} \times \mathbf{y})\).
(a) \(\mathbf{x}=(1,2,-1), \mathbf{y}=(-2,3,-1)\)
(b) \(\mathbf{x}=(-2,1,4), \mathbf{y}=(3,1,2)\)
(c) \(\mathbf{x}=(1,3,-2), \mathbf{y}=(3,9,6)\)
(d) \(\mathbf{x}=(-1,4,1), \mathbf{y}=(3,2,-1)\)
- Answer
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(a) \(\mathbf{x} \times \mathbf{y}=(1,3,7)\)
(b) \(\mathbf{x} \times \mathbf{y}=(-2,16,-5)\)
(c) \(\mathbf{x} \times \mathbf{y}=(36,-12,0)\)
(d) \(\mathbf{x} \times \mathbf{y}=(-6,2,-14)\)
Exercise \(\PageIndex{2}\)
Find the area of the parallelogram in \(\mathbb{R}^3\) that has the vectors \(\mathbf{x}=(2,3,1)\) and \(\mathbf{y}=(-3,3,1)\) for adjacent sides.
Exercise \(\PageIndex{3}\)
Find the area of the parallelogram in \(\mathbb{R}^2\) that has the vectors \(\mathbf{x}=(3,1)\) and \(\mathbf{y}=(1,4)\) for adjacent sides.
- Answer
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11
Exercise \(\PageIndex{4}\)
Find the area of the parallelogram in \(\mathbb{R}^3\) that has vertices at (1, 1, 1), (2, 3, 2), (−2, 4, 4), and (−3, 2, 3).
Exercise \(\PageIndex{5}\)
Find the area of the parallelogram in \(\mathbb{R}^2\) that has vertices at (2,−1), (4,−2), (3,0), and (1,1).
- Answer
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3
Exercise \(\PageIndex{6}\)
Find the area of the triangle in \(\mathbb{R}^3\) that has vertices at (1, 1, 0), (2, 3, 1), and (−1, 3, 2).
Exercise \(\PageIndex{7}\)
Find the area of the triangle in \(\mathbb{R}^2\) that has vertices at (−1, 2), (2, −1), and (1, 3).
- Answer
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\(\frac{9}{2}\)
Exercise \(\PageIndex{8}\)
Find the volume of the parallelepiped that has the vectors \(\mathbf{x}=(1,2,1)\), \(\mathbf{y}=(-1,1,1)\), and \(\mathbf{z}=(-1,-1,6)\) for adjacent sides.
Exercise \(\PageIndex{9}\)
A parallelepiped has base vertices at (1, 1, 1), (2, 3, 2), (−2, 4, 4), and (−3, 2, 3) and top vertices at (2, 2, 6), (3, 4, 7), (−1, 5, 9), and (−2, 3, 8). Find its volume.
- Answer
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42
Exercise \(\PageIndex{10}\)
Verify the properties of the cross product stated in Equation (1.3.8) through (1.3.12).
Exercise \(\PageIndex{11}\)
Since \(\vert \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y}) \vert\) \(\vert \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) \vert \), and \(\vert \mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) \vert \) are all equal to the volume of a parallelepiped with adjacent edges \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\), they should all have the same value.
Show that in fact
\[ \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y}) = \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) = \mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) \nonumber \]
How do these compare with \( \mathbf{z} \cdot (\mathbf{y} \times \mathbf{z})\), \(\mathbf{y} \cdot (\mathbf{z} \times \mathbf{x})\), and \(\mathbf{x} \cdot (\mathbf{z} \times \mathbf{y} ) \)?
Exercise \(\PageIndex{12}\)
Suppose \(\mathbf{x}\) and \(\mathbf{y}\) are parallel vectors in\(\mathbb{R}^3\). Show directly from the definition of the cross product that \(\mathbf{x} \times \mathbf{y}=\mathbf{0}\).
Exercise \(\PageIndex{13}\)
Show by example that the cross product is not associative. That is, find vectors \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) such that \[\mathbf{x} \times (\mathbf{y} \times \mathbf{z}) \neq (\mathbf{x} \times \mathbf{y}) \times \mathbf{z} \nonumber . \]
- Answer
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For example, \(\mathbf{e}_{2} \times\left(\mathbf{e}_{2} \times \mathbf{e}_{3}\right)=-\mathbf{e}_{3}\), whereas \(\left(\mathbf{e}_{2} \times \mathbf{e}_{2}\right) \times \mathbf{e}_{3}=\mathbf{0}\).