# 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 $$\PageIndex{1}$$

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

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

Sketches will vary slightly depending on orientation.

##### Exercise $$\PageIndex{2}$$

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

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

Sketches will vary slightly depending on orientation.

##### Exercise $$\PageIndex{3}$$

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

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

Sketches will vary slightly depending on orientation.

##### Exercise $$\PageIndex{4}$$

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

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

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.

##### Exercise $$\PageIndex{5}$$

Sketches may vary slightly.

##### Exercise $$\PageIndex{6}$$

Sketches may vary slightly.

##### Exercise $$\PageIndex{7}$$

Sketches may vary slightly.

##### Exercise $$\PageIndex{8}$$

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.

##### Exercise $$\PageIndex{9}$$

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

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

##### Exercise $$\PageIndex{10}$$

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

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

##### Exercise $$\PageIndex{11}$$

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

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

##### Exercise $$\PageIndex{12}$$

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

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