1.1.E: Introduction to Rⁿ (Exercises)

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Sep 2, 2021
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- 1.1: Introduction to Rⁿ
- 1.2: Angles and The Dot Product

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Exercise $\PageIndex{1}$

Let $\mathbf{x}=(1,2), \mathbf{y}=(2,3),$ and $\mathbf{z}=(-2,4) .$ For each of the following, plot the points $\mathbf{x}, \mathbf{y}, \mathbf{z},$ and the indicated point $\mathbf{w}$ .
(a) $\mathbf{w}=\mathbf{x}+\mathbf{y}$
(b) $\mathbf{w}=2 \mathbf{x}-\mathbf{y}$
(c) $\mathbf{w}=\mathbf{z}-2 \mathbf{x}$
(d) $\mathbf{w}=3 \mathbf{x}+2 \mathbf{y}-\mathbf{z}$

Exercise $\PageIndex{2}$

Let $\mathbf{x}=(1,3,-1), \mathbf{y}=(3,2,1),$ and $\mathbf{z}=(-2,4,-2) .$ Compute each of the following.
(a) $\mathbf{x}+\mathbf{y}$
(b) $\mathbf{x}-\mathbf{z}+3 \mathbf{y}$
(c) $3 \mathbf{z}-2 \mathbf{y}$
(d) $-3 \mathbf{x}+4 \mathbf{z}$

Answer

(a) (4,5,0)

(b) (12,5,4)

(d) (-11,7,-5)

Exercise $\PageIndex{3}$

Let $\mathbf{x}=(1,-1,2,3), \mathbf{y}=(-2,3,1,-2),$ and $\mathbf{z}=(2,1,3,-4) .$ Compute each of the following.
(a) $\mathbf{x}-2 \mathbf{z}$
(b) $\mathbf{y}+\mathbf{x}-3 \mathbf{z}$
(c) $-3 \mathbf{y}-\mathbf{x}+4 \mathbf{z}$
(d) $\mathbf{x}+3 \mathbf{z}-4 \mathbf{y}$

Exercise $\PageIndex{4}$

Let $\mathbf{x}=(1,2)$ and $\mathbf{y}=(-2,3) .$ Compute each of the following.
(a) $\|\mathbf{x}\|$
(b) $\|\mathbf{x}-\mathbf{y}\|$
(c) $\|3 \mathbf{x}\|$
(d) $\|-4 \mathbf{y}\|$

Exercise $\PageIndex{5}$

Let $x=(2,3,-1), y=(2,-1,5),$ and $z=(3,-1,-2) .$ Compute each of the following.
(a) $\|\mathbf{x}\|$
(b) $\|\mathbf{x}+2 \mathbf{y}\|$
(c) $\|-5 \mathbf{x}\|$
(d) $\|\mathbf{x}+\mathbf{y}+\mathbf{z}\|$

Answer

(a) $\sqrt{14}$

(b) $\sqrt{118}$

(d) $3 \sqrt{6}$

Exercise $\PageIndex{6}$

Find the distances between the following pairs of points.
(a) $\mathbf{x}=(3,2), \mathbf{y}=(-1,3)$
(b) $\mathbf{x}=(1,2,1), \mathbf{y}=(-2,-1,3)$
(c) $\mathbf{x}=(4,2,1,-1), \mathbf{y}=(1,3,2,-2)$
(d) $\mathbf{z}=(3,-3,0), \mathbf{y}=(-1,2,-5)$
(e) $\mathbf{w}=(1,2,4,-2,3,-1), \mathbf{u}=(3,2,1,-3,2,1)$

Answer

(a) $\sqrt{17}$

(b) $\sqrt{22}$

(d) $\sqrt{66}$

(e) $\sqrt{19}$

Exercise $\PageIndex{7}$

Draw a picture of the following sets of points in $\mathrm{R}^{2}$ .
(a) $S^{1}((1,2), 1)$
(b) $B^{2}((1,2), 1)$
(c) $\overline{B}^{2}((1,2), 1)$

Exercise $\PageIndex{8}$

Draw a picture of the following sets of points in $\mathbb{R}$ .
(a) $S^{0}(1,3)$
(b) $B^{1}(1,3)$
(c) $\overline{B}^{1}(1,3)$

Exercise $\PageIndex{9}$

Describe the differences between $S^{2}((1,2,1), 1), B^{3}((1,2,1), 1),$ and $\overline{B}^{3}((1,2,1), 1)$ in $\mathbb{R}^{3}$ .

Exercise $\PageIndex{10}$

Is the point $(1,4,5)$ in the the open ball $B^{3}((-1,2,3), 4) ?$

Exercise $\PageIndex{11}$

Is the point $(3,2,-1,4,1)$ in the open ball $B^{5}((1,2,-4,2,3), 3) ?$

Answer: No

Exercise $\PageIndex{12}$

Find the length and direction of the following vectors.
(a) $\mathbf{x}=(2,1)$
(b) $\mathbf{z}=(1,1,-1)$
(c) $\mathbf{x}=(-1,2,3)$
(d) $\mathbf{w}=(1,-1,2,-3)$

Answer

(a) $\|\mathbf{x}\|=\sqrt{5}$ , Direction: $\|\mathbf{u}\|=\frac{1}{\sqrt{5}}(2,1)$

(b) $\|\mathbf{z}\|=\sqrt{3}$ , Direction: $\|\mathbf{u}\|=\frac{1}{\sqrt{3}}(1,1,-1)$

(d) $\|\mathbf{w}\|=\sqrt{15}$ , Direction: $\|\mathbf{u}\|=\frac{1}{\sqrt{15}}(1,-1,2,-3)$

Exercise $\PageIndex{13}$

Let $\mathbf{x}=(1,3), \mathbf{y}=(4,1),$ and $\mathbf{z}=(2,-1) .$ Plot $\mathbf{x}, \mathbf{y},$ and $\mathbf{z} .$ Also, show how to obtain each of the following geometrically.
(a) $\mathbf{w}=\mathbf{x}+\mathbf{y}$
(b) $\mathbf{w}=\mathbf{y}-\mathbf{x}$
(c) $\mathbf{w}=3 \mathbf{z}$
(d) $\mathbf{w}=-2 \mathbf{z}$
(e) $\mathbf{w}=\frac{1}{2} \mathrm{z}$
(f) $\mathbf{w}=\mathbf{x}+\mathbf{y}+\mathbf{z}$
(g) $\mathbf{w}=\mathbf{x}+3 \mathbf{z}$
(h) $\mathbf{w}=\mathbf{x}-\frac{1}{4} \mathbf{y}$

Exercise $\PageIndex{14}$

Suppose $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right), \mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right),$ and $\mathbf{z}=\left(z_{1}, z_{2}, \ldots, z_{n}\right)$ are vectors in $\mathbb{R}^{n}$ and $a, b,$ and $c$ are scalars. Verify the following.
(a) $\mathbf{x}+\mathbf{y}=\mathbf{y}+\mathbf{x}$
(b) $\mathbf{x}+(\mathbf{y}+\mathbf{z})=(\mathbf{x}+\mathbf{y})+\mathbf{z}$
(c) $a(\mathbf{x}+\mathrm{y})=a \mathrm{x}+a \mathrm{y}$
(d) $(a+b) \mathbf{x}=a \mathbf{x}+b \mathbf{x}$
(e) $a(b \mathbf{x})=(a b) \mathbf{x}$
(f) $\mathbf{x}+\mathbf{0}=\mathbf{x}$
(g) $1 \mathbf{x}=\mathbf{x}$
(h) $\mathbf{x}+(-\mathbf{x})=0,$ where $-\mathbf{x}=-1 \mathbf{x}$

Exercise $\PageIndex{15}$

Let $\mathbf{u}=(1,1)$ and $\mathbf{v}=(-1,1)$ be vectors in $\mathbb{R}^{2}$
(a) Let $\mathbf{x}=(2,1) .$ Find scalars $a$ and $b$ such that $\mathbf{x}=a \mathbf{u}+b \mathbf{v} .$ Are $a$ and $b$ unique?
(b) Let $\mathbf{x}=(x, y)$ be an arbitrary vector in $\mathbb{R}^{2} .$ Show that there exist unique scalars $a$ and $b$ such that $\mathbf{x}=a \mathbf{u}+b \mathbf{v}$ .
(c) The result in (b) shows that u and v form a basis for $\mathbb{R}^{2}$ which is different from the standard basis of $\mathbf{e}_{1}$ and $\mathbf{e}_{2} .$ Show that the vectors $\mathbf{u}=(1,1)$ and $\mathbf{w}=(-1,-1)$ do not form a basis for $\mathbb{R}^{2} .$ (Hint: Show that there do not exist scalars $a$ and $b$ such that $\mathbf{x}=a \mathbf{u}+\mathbf{w} \text { when } \mathbf{x}=(2,1).)$

Answer

(a) $a=\frac{3}{2}, b=-\frac{1}{2}$ ; Yes, $a$ and $b$ are unique.

(b) $a=\frac{x+y}{2}, b=\frac{y-x}{2}$

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