5.3.1: Exercises 5.3
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In Exercises \(\PageIndex{1}\) – \(\PageIndex{4}\), vectors \(\vec{x}\) and \(\vec{y}\) are given. Sketch \(\vec{x}\), \(\vec{y}\), \(\vec{x}+\vec{y}\), and \(\vec{x}-\vec{y}\) on the same Cartesian axes.
\(\vec{x}=\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{2}\\{3}\\{2}\end{array}\right]\)
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\(\vec{x}+\vec{y}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{-1}\\{-4}\\{0}\end{array}\right]\)
Sketches will vary slightly depending on orientation.
\(\vec{x}=\left[\begin{array}{c}{2}\\{4}\\{-1}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{-1}\\{-3}\\{-1}\end{array}\right]\)
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\(\vec{x}+\vec{y}=\left[\begin{array}{c}{1}\\{1}\\{-2}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{3}\\{7}\\{0}\end{array}\right]\)
Sketches will vary slightly depending on orientation.
\(\vec{x}=\left[\begin{array}{c}{1}\\{1}\\{2}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{3}\\{3}\\{6}\end{array}\right]\)
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\(\vec{x}+\vec{y}=\left[\begin{array}{c}{4}\\{4}\\{8}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{-2}\\{-2}\\{-4}\end{array}\right]\)
Sketches will vary slightly depending on orientation.
\(\vec{x}=\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{0}\\{-1}\\{1}\end{array}\right]\)
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\(\vec{x}+\vec{y}=\left[\begin{array}{c}{0}\\{0}\\{2}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{0}\\{2}\\{0}\end{array}\right]\)
Sketches may vary slightly.
In Exercises \(\PageIndex{5}\) - \(\PageIndex{8}\), vectors \(\vec{x}\) and \(\vec{y}\) are drawn. Sketch \(2\vec{x}\), \(-\vec{y}\), \(\vec{x}+\vec{y}\), and \(\vec{x}-\vec{y}\) on the same Cartesian axes.
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Sketches may vary slightly.
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Sketches may vary slightly.
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Sketches may vary slightly.
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Sketches may vary slightly.
In Exercises \(\PageIndex{9}\) - \(\PageIndex{12}\), a vector \(\vec{x}\) and a scalar \(a\) are given. Using Definition 3D Vector Length, compute the lengths of \(\vec{x}\) and \(a\vec{x}\), then compare these lengths.
\(\vec{x}=\left[\begin{array}{c}{1}\\{-2}\\{5}\end{array}\right],\: a=2\)
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\(||\vec{x}||=\sqrt{30}\), \(||a\vec{x}||=\sqrt{120}=2\sqrt{30}\)
\(\vec{x}=\left[\begin{array}{c}{-3}\\{4}\\{3}\end{array}\right],\: a=-1\)
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\(||\vec{x}||=\sqrt{34}\), \(||a\vec{x}||=\sqrt{34}\)
\(\vec{x}=\left[\begin{array}{c}{7}\\{2}\\{1}\end{array}\right],\: a=5\)
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\(||\vec{x}||=\sqrt{54}=3\sqrt{6}\), \(||a\vec{x}||=\sqrt{270}=15\sqrt{6}\)
\(\vec{x}=\left[\begin{array}{c}{1}\\{2}\\{-2}\end{array}\right],\: a=3\)
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\(||\vec{x}||=\sqrt{3}\), \(||a\vec{x}||=\sqrt{27}\)