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.

Exercise $\PageIndex{1}$

$\vec{x}=\left[\begin{array}{c}{1}\\{-1}\\{2}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{2}\\{3}\\{2}\end{array}\right]$

Answer

$\vec{x}+\vec{y}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{-1}\\{-4}\\{0}\end{array}\right]$

Sketches will vary slightly depending on orientation.

$clipboard_efebe63539871aaf98d49f0e94b585d02.png$

Exercise $\PageIndex{2}$

$\vec{x}=\left[\begin{array}{c}{2}\\{4}\\{-1}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{-1}\\{-3}\\{-1}\end{array}\right]$

Answer

$\vec{x}+\vec{y}=\left[\begin{array}{c}{1}\\{1}\\{-2}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{3}\\{7}\\{0}\end{array}\right]$

Sketches will vary slightly depending on orientation.

$clipboard_e49bfdcb2cdf563fce7a6726a5e827a37.png$

Exercise $\PageIndex{3}$

$\vec{x}=\left[\begin{array}{c}{1}\\{1}\\{2}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{3}\\{3}\\{6}\end{array}\right]$

Answer

$\vec{x}+\vec{y}=\left[\begin{array}{c}{4}\\{4}\\{8}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{-2}\\{-2}\\{-4}\end{array}\right]$

Sketches will vary slightly depending on orientation.

$clipboard_e779d4589a35f4316924d5a02c6b66f1d.png$

Exercise $\PageIndex{4}$

$\vec{x}=\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right],\:\vec{y}=\left[\begin{array}{c}{0}\\{-1}\\{1}\end{array}\right]$

Answer

$\vec{x}+\vec{y}=\left[\begin{array}{c}{0}\\{0}\\{2}\end{array}\right],\:\vec{x}-\vec{y}=\left[\begin{array}{c}{0}\\{2}\\{0}\end{array}\right]$

Sketches may vary slightly.

$clipboard_ea67d351f37d44bc7e925b992703b86fa.png$

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.

Exercise $\PageIndex{5}$

$clipboard_e0e4a8a1893fd49951e7da46cb94e11f4.png$

Answer

Sketches may vary slightly.

$clipboard_efa79a08d2158bbde48be54acf10988bc.png$

Exercise $\PageIndex{6}$

$clipboard_e7f39b8b79723f5ae5bef0d45f7e40c50.png$

Answer

Sketches may vary slightly.

$clipboard_eb94719cae9ad39ce67c10de08d86dab6.png$

Exercise $\PageIndex{7}$

$clipboard_eb68625311a3f29ff2eda8177ca7ad091.png$

Answer

Sketches may vary slightly.

$clipboard_e07a3fcd612e33589067c4a347f8ea952.png$

Exercise $\PageIndex{8}$

$clipboard_e60a8499a8f311fc8a3ed9db72cc1acf1.png$

Answer

Sketches may vary slightly.

$clipboard_e9c03932841e2fec8a42c08a09e98b7d7.png$

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.

Exercise $\PageIndex{9}$

$\vec{x}=\left[\begin{array}{c}{1}\\{-2}\\{5}\end{array}\right],\: a=2$

Answer: $||\vec{x}||=\sqrt{30}$, $||a\vec{x}||=\sqrt{120}=2\sqrt{30}$

Exercise $\PageIndex{10}$

$\vec{x}=\left[\begin{array}{c}{-3}\\{4}\\{3}\end{array}\right],\: a=-1$

Answer: $||\vec{x}||=\sqrt{34}$, $||a\vec{x}||=\sqrt{34}$

Exercise $\PageIndex{11}$

$\vec{x}=\left[\begin{array}{c}{7}\\{2}\\{1}\end{array}\right],\: a=5$

Answer: $||\vec{x}||=\sqrt{54}=3\sqrt{6}$, $||a\vec{x}||=\sqrt{270}=15\sqrt{6}$

Exercise $\PageIndex{12}$

$\vec{x}=\left[\begin{array}{c}{1}\\{2}\\{-2}\end{array}\right],\: a=3$

Answer: $||\vec{x}||=\sqrt{3}$, $||a\vec{x}||=\sqrt{27}$

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