Math Problems Vault

Problem 51.1

(a)(b)

Find necessary and sufficient conditions for a given set of $\binom {n}{2}$ nonnegative real numbers to serve as the set of distances between:

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(a)

$n$ affinely independent points of a Euclidean space;

(b)

$n$ arbitrary points of a Euclidean space.

Problem 51.2

(a)(b)(c)(d)

Does there exist, in a Euclidean space, a set of points $a_{1}$ , $a_{2}, a_{3}, a_{4}, a_{5}$ for which the matrix $A$ is the matrix of distances $(\rho \left(a_{i}, a_{j}\right))$ , and what is the smallest dimension of a space into which such a set can be placed:

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(a)

$A=\left(\begin{array}{ccccc}0 & 1 & 2 & 2 & 2 \sqrt{2} \\ 1 & 0 & \sqrt{5} & \sqrt{5} & 3 \\ 2 & \sqrt{5} & 0 & 2 \sqrt{2} & 2 \\ 2 & \sqrt{5} & 2 \sqrt{2} & 0 & 2 \sqrt{3} \\ 2 \sqrt{2} & 3 & 2 & 2 \sqrt{3} & 0\end{array}\right)$ ;

(b)

$A=\left(\begin{array}{ccccc}0 & 3 & \sqrt{5} & \sqrt{5} & 2 \sqrt{2} \\ 3 & 0 & \sqrt{14} & \sqrt{14} & \sqrt{17} \\ \sqrt{5} & \sqrt{14} & 0 & \sqrt{2} & \sqrt{17} \\ \sqrt{5} & \sqrt{14} & \sqrt{2} & 0 & 3 \\ 2 \sqrt{2} & \sqrt{17} & \sqrt{17} & 3 & 0\end{array}\right)$ ;

(c)

$A=\left(\begin{array}{ccccc}0 & 1 & 2 & \sqrt{5} & 1 \\ 1 & 0 & \sqrt{5} & 2 & \sqrt{2} \\ 2 & \sqrt{5} & 0 & \sqrt{17} & 1 \\ \sqrt{5} & 2 & \sqrt{17} & 0 & \sqrt{10} \\ 1 & \sqrt{2} & 1 & \sqrt{10} & 0\end{array}\right)$ ;

(d)

$A=\left(\begin{array}{ccccc}0 & \sqrt{5} & \sqrt{5} & \sqrt{5} & \sqrt{5} \\ \sqrt{5} & 0 & 2 \sqrt{5} & 2 \sqrt{2} & 2 \\ \sqrt{5} & 2 \sqrt{5} & 0 & 2 & 2 \sqrt{2} \\ \sqrt{5} & 2 \sqrt{2} & 2 & 0 & 2 \sqrt{5} \\ \sqrt{5} & 2 & 2 \sqrt{2} & 2 \sqrt{5} & 0\end{array}\right)$ .

Problem 51.3

(a)(b)

Prove the equivalence of the following two properties of a pair of flats $\left\{ P, Q\right\}$ in a Euclidean space:

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(a)

any line lying in one of these flats is perpendicular to any line lying in the other flat;

(b)

the flats $P, Q$ are perpendicular and either skew or intersect in a single point.

Problem 51.4

Let $Q \subset P$ be flats in an $n$ -dimensional Euclidean space $E$ . Prove that any flat $P^{\prime } \subset E$ perpendicular to $P$ , for which $P \cap P^{\prime }=Q$ , has dimension $\leqslant n-\operatorname {dim} P+\operatorname {dim} Q$ , and that there is a unique such flat of dimension $n-\operatorname {dim} P+\operatorname {dim} Q$ .

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Problem 51.5

(a)(b)(c)

Let $P$ be a flat in a Euclidean space and let the point $a$ not belong to $P$ . Prove that:

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(a)

there is a unique line passing through the point $a$ , meeting $P$ and perpendicular to $P$ ;

(b)

if $c$ is any point in $P$ and $z$ is the orthogonal component of the vector $\overline{a c}$ with respect to the directing subspace of the flat $P$ , then $a+\langle z\rangle$ is the line indicated in (a); $a+z$ is the point of intersection of this line with $P$ ;

(c)

$\rho (a, P)=\left|z\right|$ .

Problem 51.6

(a)(b)

In a Euclidean space find the line passing through the point $a$ , meeting the flat $P$ and perpendicular to $P$ , if:

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(a)

$a=(5,-4,4,0), \quad P=(2,-1,2,3)+\langle (1,1,1,2),(2,2,1,1)\rangle$ ;

(b)

$a=(5,0,2,11), \quad P: \begin{cases} x_{1}+5 x_{2}+x_{4}=10, \\ 5 x_{1}+x_{2}+3 x_{3}+8 x_{4}=-1. \end{cases}$

Problem 51.7

(a)(b)(c)(d)

In a Euclidean space find the distance from the point $a$ to the flat $P$ , if:

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(a)

$a=(4,1,-4,-5), \quad P=(3,-2,1,5)+\langle (2,3,-2,-2),(4,1,3,2)\rangle$ ;

(b)

$a=(1,1,-2,-3,-2), \quad P=(3,7,-5,4,1)+\langle (1,1,2,0,1),(2,2,1,3,1)\rangle ;$

(c)

$a=(2,1,-3,4), \quad P: \begin{cases} 2 x_{1}-4 x_{2}-8 x_{3}+13 x_{4}=-19, \\ x_{1}+x_{2}-x_{3}+2 x_{4}=1 ; \end{cases}$

(d)

$a=(1,-3,-2,9,-4), \quad P: \begin{cases} x_{1}-2 x_{2}-3 x_{3}+3 x_{4}+2 x_{5}=-2, \\ x_{1}-2 x_{2}-7 x_{3}+5 x_{4}+3 x_{5}=1. \end{cases}$

Problem 51.8

In an $n$ -dimensional Euclidean space find the distance from the point $\left(b_{1}, \ldots , b_{n}\right)$ to the hyperplane $\sum_{i=1}^{n} a_{i} x_{i}=c$ .

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Problem 51.9

In the space of polynomials with scalar product $(f, g)=\int_{-1}^{1} f(x) g(x) d x$ find the distance from the polynomial $x^{n}$ to the subspace of polynomials of degree less than $n$ .

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Problem 51.10

In the space of trigonometric polynomials with scalar product $(f, g)=\int_{-\pi }^{\pi } f(x) g(x) d x$ find the distance from the function $\cos^{n+1} x$ to the subspace

\langle 1, \cos x, \sin x, \ldots , \cos n x, \sin n x\rangle

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Problem 51.11

Let $P$ be a flat in an $n$ -dimensional Euclidean space $E$ . Prove that through every point $a \in E$ there passes a unique flat $Q$ of dimension $n-\operatorname {dim} P$ , perpendicular to $P$ and meeting it in a single point.

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Problem 51.12

(a)(b)

Find, in a Euclidean space, the flat of largest dimension passing through the point $a$ , perpendicular to the flat $P$ and meeting it in a single point, if:

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(a)

$a=(2,-1,3,5), \quad P=(7,2,-3,4)+\langle (-1,3,2,1),(1,2,3,-1)\rangle$ ;

(b)

$a=(3,-2,1,4), \quad P: \begin{cases} 2 x_{1}+3 x_{2}-x_{3}-2 x_{4}=4, \\ 3 x_{1}+2 x_{2}-5 x_{3}+x_{4}=5. \end{cases}$

Problem 51.13

(a)(b)(c)(d)

Let $P_{1}=c_{1}+L_{1}$ and $P_{2}=c_{2}+L_{2}$ be two disjoint flats in a Euclidean space, let $y$ and $z$ be respectively the orthogonal projection and the orthogonal component of the vector $\overline{c_{1} c_{2}}$ with respect to the subspace $L_{1}+L_{2}$ , and let $y=y_{1}+y_{2}$ , where $y_{1} \in L_{1}$ , $y_{2} \in L_{2}$ .

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(a)

Prove that the line $c_{1}+y_{1}+\langle z\rangle$ is perpendicular to the flats $P_{1}, P_{2}$ and meets $P_{1}$ at the point $c_{1}+y_{1}$ , and $P_{2}$ at the point $c_{2}-y_{2}$ .

(b)

Find the distance $\rho \left(P_{1}, P_{2}\right)$ .

(c)

Establish a bijective correspondence between $L_{1} \cap L_{2}$ and the set of all lines perpendicular to $P_{1}$ and $P_{2}$ and meeting both flats.

(d)

Show that all the lines described in (c) are parallel to each other and that their union is a flat of dimension $\operatorname {dim}\left(L_{1} \cap L_{2}\right)+1$ .

Problem 51.14

(a)(b)(c)(d)

In a Euclidean space find the distance between the flats $P_{1}$ and $P_{2}$ , if:

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(a)

$P_{1}: \begin{cases} x_{1}+3 x_{2}+x_{3}+x_{4}=3, \\ x_{1}+3 x_{2}-x_{3}+2 x_{4}=6, \end{cases}$ $P_{2}=(0,2,6,-5)+\langle (-7,1,1,1),(-10,1,2,3)\rangle ;$

(b)

$P_{1}: \begin{cases} -x_{1}+x_{2}+x_{3}+x_{4}=3, \\ -3 x_{2}+2 x_{3}-4 x_{4}=4, \end{cases}$ $P_{2}=(1,3,-3,-1)+\langle (1,0,1,1)\rangle ;$

(c)

$P_{1}: \begin{cases} x_{1}+x_{3}+x_{4}-2 x_{5}=2, \\ x_{2}+x_{3}-x_{4}-x_{5}=3, \\ x_{1}-x_{2}+2 x_{3}-x_{5}=3, \end{cases}$ $P_{2}=(1,-2,5,8,2)+\langle (0,1,2,1,2),(2,1,2,-1,1)\rangle ;$

(d)

$P_{1}: \begin{cases} x_{1}-2 x_{2}+x_{3}-x_{4}+3 x_{5}=6, \\ x_{1}-x_{3}-x_{4}+3 x_{5}=0, \end{cases}$ $P_{2}=(-4,3,-3,2,4)+\langle (2,0,1,1,1),(-5,1,0,1,1)\rangle$ .

Problem 51.15

Points $a_{0}, a_{1}, \ldots , a_{n}$ in a Euclidean space are located at equal distance $d$ from each other. Find the distance between the flats $\left\langle a_{0}, a_{1}, \ldots , a_{k}\right\rangle$ and $\left\langle a_{k+1}, \ldots , a_{n}\right\rangle$ .

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Problem 51.16

Prove that configurations of two flats

\left\{ a_{1}+L_{1}, a_{2}+L_{2}\right\} \quad \text{ and } \quad \left\{ a_{1}^{\prime }+L_{1}^{\prime }, a_{2}^{\prime }+L_{2}^{\prime }\right\}

in a Euclidean space are metrically congruent if and only if $\rho \left(a_{1}+L_{1}, a_{2}+L_{2}\right)=\rho \left(a_{1}^{\prime }+L_{1}^{\prime }, a_{2}^{\prime }+L_{2}^{\prime }\right)$ , and the configurations of subspaces $L_{1}, L_{2}, L_{1}^{\prime }, L_{2}^{\prime }$ are orthogonally congruent in the corresponding Euclidean vector space.

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Problem 51.17

(a)(b)(c)

Determine whether the given pairs of flats are metrically congruent in a Euclidean space:

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(a)

$P_{1}=(0,9,8,-12,11)+\langle (0,2,2,2,1),(3,1,1,1,-1)\rangle$ , $P_{2}=(-3,-4,-5,11,-12)+\langle (7,5,-5,-1,-5),(3,5,-1,11,13)\rangle ;$

(b)

$Q_{1}=(2,-5,-11,-8,-10)+\langle (2,-1,1,-1,1),(2,-2,1,0,1)\rangle$ , $Q_{2}=(8,8,10,9,11)+\langle (0,3,4,-4,-3),(14,-2,-5,3,4)\rangle ;$

(c)

$R_{1}=(7,-3,-9,-14,5)+\langle (0,0,0,1,2),(2,-1,2,0,-6)\rangle$ , $R_{2}=(0,10,9,14,-5)+\langle (1,7,2,0,6),(4,-1,0,2,-2)\rangle$ .

Problem 51.18

(a)(b)

Find, in a Euclidean space, the locus of points through which a line can be drawn meeting the flats $P_{1}, P_{2}$ and perpendicular to these flats:

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(a)

$P_{1}=(1,2,-1,-9,-13)+\langle (2,3,7,10,13),(3,5,11,16,21)\rangle$ , $P_{2}: \begin{cases} 3 x_{1}-5 x_{2}+2 x_{3}-x_{4}+x_{5}=-22, \\ 2 x_{1}+4 x_{2}+3 x_{3}-x_{4}-3 x_{5}=-4, \\ 9 x_{1}+3 x_{2}+x_{3}-2 x_{4}-2 x_{5}=-138 ; \end{cases}$

(b)

$P_{1}=(3,7,2,4,-3)+\langle (2,5,4,5,3),(4,5,6,3,3)\rangle$ , $P_{2}: \begin{cases} -3 x_{1}+2 x_{2}+x_{3}-2 x_{4}+x_{5}=-14, \\ 6 x_{1}-x_{2}-4 x_{3}+2 x_{4}-x_{5}=16, \\ 2 x_{1}-x_{2}+2 x_{4}-3 x_{5}=26. \end{cases}$

Problem 51.19

(a)(b)

Prove that:

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(a)

if a motion of a Euclidean space has two skew invariant flats, then it has a fixed point;

(b)

a motion $f$ of an $n$ -dimensional Euclidean space having a fixed point has two skew invariant flats of positive dimension, if $f$ is proper, $n \geqslant 5$ , and odd, or $f$ is improper, $n \geqslant 4$ , and even.

Problem 51.20

Let $a_{0}, a_{1}, \ldots , a_{s}$ and $b_{0}, b_{1}, \ldots , b_{s}$ be two sets of points in a Euclidean space. Prove that a motion taking each point $a_{i}$ to the point $b_{i}$ exists if and only if $\rho \left(a_{i}, a_{j}\right)=\rho \left(b_{i}, b_{j}\right)(i, j=1, \ldots , s)$ .

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Problem 51.21

Prove that for every motion $f$ of a Euclidean space, the set of points $a$ at which the minimum of the distance $\rho (a, f(a))$ is attained is a flat invariant under $f$ , and that the restriction of $f$ to this flat is a translation.

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Problem 51.22

Prove that if, for two tetrahedra in a three-dimensional Euclidean space, the corresponding dihedral angles are equal, then these tetrahedra are similar.

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Problem 51.23

Give a geometric description of the proper motion $f$ of Euclidean space, if:

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(a)

$D f=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right), \quad f(O)=(-2,4) ;$

(b)

$D f=\frac{1}{\sqrt{2}}\left(\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right), \quad f(O)=(1,1)$ ;

(c)

$D f=\frac{1}{3}\left(\begin{array}{rrr}2 & -1 & 2 \\ 2 & 2 & -1 \\ -1 & 2 & 2\end{array}\right), \quad f(O)=(1,0,-1)$ ;

(d)

$D f=\frac{1}{9}\left(\begin{array}{rrr}4 & 1 & -8 \\ 7 & 4 & 4 \\ 4 & -8 & 1\end{array}\right), \quad f(O)=(-1,-7,2)$ ;

(e)

$D f=\frac{1}{7}\left(\begin{array}{rrr}-2 & 3 & 6 \\ 6 & -2 & 3 \\ 3 & 6 & -2\end{array}\right), \quad f(O)=(-2,4,1)$ .

Problem 51.24

Give a geometric description of the improper motion $f$ of Euclidean space, if:

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(a)

$D f=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), \quad f(O)=(1,0)$ ;

(b)

$D f=\frac{1}{2}\left(\begin{array}{cc}1 & \sqrt{3} \\ \sqrt{3} & -1\end{array}\right), \quad f(O)=(1,-\sqrt{3})$ .

(c)

$D f=-\frac{1}{9}\left(\begin{array}{rrr}4 & 1 & -8 \\ 7 & 4 & 4 \\ 4 & -8 & 1\end{array}\right), \quad f(O)=(1,1,-2)$ ;

(d)

$D f=\frac{1}{3}\left(\begin{array}{rrr}2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2\end{array}\right), \quad f(O)=(4,0,2)$ ;

(e)

$D f=\frac{1}{3}\left(\begin{array}{rrr}-1 & 2 & 2 \\ -2 & 1 & -2 \\ 2 & 2 & -1\end{array}\right), \quad f(O)=(2,0,0)$ ;

(f)

$D f=\frac{1}{7}\left(\begin{array}{rrr}-2 & 3 & 6 \\ 3 & 6 & -2 \\ 6 & -2 & 3\end{array}\right), \quad f(O)=(-3,1,2)$ .