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1.1.E: Introduction to Rⁿ (Exercises)

  • Page ID
    31488
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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)

    (c) (-12,8,-8)

    (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) \(\sqrt{118}\)

    (c) \(5 \sqrt{14}\)

    (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) \(\sqrt{22}\)

    (c) \(2 \sqrt{3}\)

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

    (c) \(\|\mathbf{x}\|=\sqrt{14}\), Direction: \(\|\mathbf{u}\|=\frac{1}{\sqrt{14}}(-1,2,3)\)

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


    This page titled 1.1.E: Introduction to Rⁿ (Exercises) is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.