Chapter 13

Affine and point Euclidean spaces

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Section 33Affine spaces[44/52%]
Problem 33.1

Verify that an nn-dimensional vector space L\mathcal{L} is an affine space with the space of vectors coinciding with L\mathcal{L}, if the points of this affine space are taken to be the vectors from L\mathcal{L}, and to every ordered pair of vectors a,ba, b there is assigned the vector x=b−ax = b - a.

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Problem 33.2

Prove that in an affine space A\mathcal{A}:

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(1)

AA→=\overrightarrow {AA} = o for any point AA from A\mathcal{A}

(2)

P(A,o)=AP(A, o) = A for any point AA from A\mathcal{A}

(3)

AB→=−BA→\overrightarrow {AB} = - \overrightarrow {BA} for any points AA and BB from A\mathcal{A}

(4)

the equality AB→=A1B1‾\overrightarrow {AB} = \overline{A_{1} B_{1}} holds if and only if AA1→=BB1→\overrightarrow {AA_{1}} = \overrightarrow {BB_{1}}.

Problem 33.3
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(1)

Prove that a system of points A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k} of an affine space is independent if and only if there does not exist a plane of dimension less than kk containing this system of points.

(2)

Prove that a system of points A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k} of an affine space is independent if and only if, for an arbitrary point OO, the equalities

λ0OA0→+λ1OA1→+…+λkOAk→=o,λ0+λ1+…+λk=0 \begin{gathered} \lambda _{0} \overrightarrow {OA_{0}} + \lambda _{1} \overrightarrow {OA_{1}} + \ldots + \lambda _{k} \overrightarrow {OA_{k}} = o, \\ \lambda _{0} + \lambda _{1} + \ldots + \lambda _{k} = 0 \end{gathered}

imply that λ0=λ1=…=λk=0\lambda_{0} = \lambda_{1} = \ldots = \lambda_{k} = 0.

Problem 33.4

Is the system of points with coordinates independent:

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(0,1,1),(1,0,1),(1,1,0)(0, 1, 1), (1, 0, 1), (1, 1, 0)

(2)

(0,1,1),(1,0,1),(1,1,0),(2,2,2)(0, 1, 1), \quad (1, 0, 1), \quad (1, 1, 0), \quad (2, 2, 2)

(3)

(0,1,1),(1,0,1),(1,1,0),(2/3,2/3,2/3)(0, 1, 1), (1, 0, 1), \quad (1, 1, 0), \quad (2 / 3, 2 / 3, 2 / 3)?

Problem 33.5

Show that the notion of independence of a system of points A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k} is symmetric with respect to all the points of this system. Namely, if the system of vectors A0A1→,A0A2→,…,A0Ak→\overrightarrow {A_{0} A_{1}}, \overrightarrow {A_{0} A_{2}}, \ldots , \overrightarrow {A_{0} A_{k}} is linearly independent, then any system AjA0→,…,AjAj−1→,AjAj+1→,…,AjAk→,j=1,2,…,k\overrightarrow {A_{j} A_{0}}, \ldots , \overrightarrow {A_{j} A_{j - 1}}, \overrightarrow {A_{j} A_{j + 1}}, \ldots , \overrightarrow {A_{j} A_{k}}, j = 1, 2, \ldots , k is also linearly independent.

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Problem 33.6

Let m\mathrm{m} and m′\mathrm{m}^{\prime } be planes with directing subspaces M\mathcal{M} and M′\mathcal{M}^{\prime }. Prove that:

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(1)

if M⊂M′\mathcal{M} \subset \mathcal{M}^{\prime }, then either m\mathrm{m} and m′\mathrm{m}^{\prime } have no common points, or m⊂m′\mathrm{m} \subset \mathrm{m}^{\prime }

(2)

if M=M′\mathcal{M} = \mathcal{M}^{\prime }, then m\mathrm{m} and m′\mathrm{m}^{\prime } either have no common points or coincide.

Problem 33.7

Prove that if a line has two distinct common points with a plane, then it belongs to that plane.

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Problem 33.8

Prove that if a kk-dimensional plane m1\mathrm{m}_{1} contains an independent system of points A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k} common with a plane m2\mathrm{m}_{2}, then m1⊂m2\mathrm{m}_{1} \subset \mathrm{m}_{2}.

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Problem 33.9

Prove that there exists exactly one kk-dimensional plane containing an independent system of points A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k}.

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Problem 33.10

Let A0,A1,…,AkA_{0}, A_{1}, \ldots , A_{k} be an independent system of points in a kk-dimensional plane m\mathrm{m}, and let O\mathrm{O} be a fixed point of the affine space. Prove that m\mathrm{m} consists of exactly those points AA for which

OA→=λ0OA0→+λ1OA1→+…+λkOAk→ \overrightarrow {OA} = \lambda _{0} \overrightarrow {OA_{0}} + \lambda _{1} \overrightarrow {OA_{1}} + \ldots + \lambda _{k} \overrightarrow {OA_{k}}

where λ0,λ1,…,λk\lambda_{0}, \lambda_{1}, \ldots , \lambda_{k} are numbers satisfying the equality λ0+λ1+…+λk=1\lambda_{0} + \lambda_{1} + \ldots + \lambda_{k} = 1.

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Problem 33.11

Let l1,l2,l3,l4l_{1}, l_{2}, l_{3}, l_{4} be lines in an affine space, with l1l_{1} parallel to l2l_{2}, and l3l_{3} parallel to l4l_{4}. Let, further, l3l_{3} intersect l1l_{1} and l2l_{2} at the points A1A_{1} and B1B_{1} respectively, and let l4l_{4} intersect l1l_{1} at the point A2A_{2}. Prove that l4l_{4} also intersects l2l_{2} at a point B2B_{2} such that A1A2→=B1B2→,A1B1→=A2B2→\overrightarrow {A_{1} A_{2}} = \overrightarrow {B_{1} B_{2}}, \overrightarrow {A_{1} B_{1}} = \overrightarrow {A_{2} B_{2}}.

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Problem 33.12

Prove that any two lines of an nn-dimensional affine space (n⩾3)(n \geqslant 3) are entirely contained in some three-dimensional plane.

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Problem 33.13

Under what necessary and sufficient condition are two lines x=a0+a1tx = a_{0} + a_{1} t and x=b0+b1tx = b_{0} + b_{1} t contained in one two-dimensional plane?

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Problem 33.14

Write the equations:

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(1)

of the line passing through the points A(−1,0,3,−2)A(- 1, 0, 3, - 2) and B(2,1,4,5)B(2, 1, 4, 5)

(2)

of the two-dimensional plane passing through the points A(−2,1,1,1),B(1,3,−5,2)A(- 2, 1, 1, 1), B(1, 3, - 5, 2) and C(0,1,1,4)C(0, 1, 1, 4)

(3)

of the three-dimensional plane (hyperplane) passing through the points A(1,1,0,−1),B(2,−1,3,3),C(1,−1,1,5)A(1, 1, 0, - 1), B(2, - 1, 3, 3), C(1, - 1, 1, 5) and D(0,0,3,−1)D(0, 0, 3, - 1).

Problem 33.15

Let A(x1′,x2′,…,xn′)A\left(x_{1}^{\prime }, x_{2}^{\prime }, \ldots , x_{n}^{\prime }\right) and B(x1′′,x2′′,…,xn′′)B\left(x_{1}^{\prime \prime }, x_{2}^{\prime \prime }, \ldots , x_{n}^{\prime \prime }\right) be two distinct points, and let pp and qq be some numbers. Find the coordinates of the point CC dividing the segment ABAB in the ratio p:qp: q.

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Problem 33.16

Let the points A,B,CA, B, C in an nn-dimensional space not lie on one line. Prove that the medians of the triangle ABCABC pass through one point and are divided at this point in the ratio 2:12: 1, counting from the vertex.

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Problem 33.17

A point MM belongs to the hyperplane given by the equation a1x1+…+anxn+a0=0a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} = 0, and the vector MM1‾\overline{MM_{1}} has coordinate column (a1,a2,…,an)T\left(a_{1}, a_{2}, \ldots , a_{n}\right)^{T}. Prove that the coordinates of the point M1M_{1} satisfy the inequality a1x1+…+anxn+a0>0a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} > 0.

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Problem 33.18

Write the parametric equations of the plane given by the system of linear equations:

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(1)

A27x=c46\mathbf{A}_{27} \mathbf{x} = \mathbf{c}_{46}. A27=[1−3−26],c46=[2−4]\mathbf{A}_{27} = \left[\begin{smallmatrix} 1 & -3 \\ -2 & 6 \end{smallmatrix}\right], \mathbf{c}_{46} = \left[\begin{smallmatrix} 2 \\ -4 \end{smallmatrix}\right].

(2)

A10x=c29\mathbf{A}_{10} \mathrm{x} = \mathbf{c}_{29}. A10=[2513],c29=[32]\mathbf{A}_{10} = \left[\begin{smallmatrix} 2 & 5 \\ 1 & 3 \end{smallmatrix}\right], \mathbf{c}_{29} = \left[\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}\right].

(3)

A198x=c123\mathbf{A}_{198} \mathbf{x} = \mathbf{c}_{123}. A198=[−286−41064−8−4],c123=[1212−8]\mathbf{A}_{198} = \left[\begin{smallmatrix} -2 & 8 & 6 \\ -4 & 10 & 6 \\ 4 & -8 & -4 \end{smallmatrix}\right], \mathbf{c}_{123} = \left[\begin{smallmatrix} 12 \\ 12 \\ -8 \end{smallmatrix}\right].

(4)

A249x=c124\mathbf{A}_{249} \mathrm{x} = \mathbf{c}_{124}. A249=[−31−26−24−15510],c124=[−36−15]\mathbf{A}_{249} = \left[\begin{smallmatrix} -3 & 1 & -2 \\ 6 & -2 & 4 \\ -15 & 5 & 10 \end{smallmatrix}\right], \mathbf{c}_{124} = \left[\begin{smallmatrix} -3 \\ 6 \\ -15 \end{smallmatrix}\right].

(5)

A267x=c66\mathbf{A}_{267} \mathbf{x} = \mathbf{c}_{66}. A267=[−13−1−35−1−331],c66=[111]\mathbf{A}_{267} = \left[\begin{smallmatrix} -1 & 3 & -1 \\ -3 & 5 & -1 \\ -3 & 3 & 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(6)

A517x=c125\mathbf{A}_{517} \mathbf{x} = \mathbf{c}_{125}. A517=[31−242−36−58−123],c125=[5313]\mathbf{A}_{517} = \left[\begin{smallmatrix} 3 & 1 & -2 & 4 \\ 2 & -3 & 6 & -5 \\ 8 & -1 & 2 & 3 \end{smallmatrix}\right], \mathbf{c}_{125} = \left[\begin{smallmatrix} 5 \\ 3 \\ 13 \end{smallmatrix}\right].

(7)

A403x=c208\mathbf{A}_{403} \mathrm{x} = \mathbf{c}_{208}. A403=[23−1475−1−2−32513],c208=[−4−2−18]\mathbf{A}_{403} = \left[\begin{smallmatrix} 2 & 3 & -1 \\ 4 & 7 & 5 \\ -1 & -2 & -3 \\ 2 & 5 & 13 \end{smallmatrix}\right], \mathbf{c}_{208} = \left[\begin{smallmatrix} -4 \\ -2 \\ -1 \\ 8 \end{smallmatrix}\right].

(8)

A586x=c123\mathbf{A}_{586} \mathbf{x} = \mathbf{c}_{123}. A586=[−33126−9−33126−92−2−8−46],c123=[1212−8]\mathbf{A}_{586} = \left[\begin{smallmatrix} -3 & 3 & 12 & 6 & -9 \\ -3 & 3 & 12 & 6 & -9 \\ 2 & -2 & -8 & -4 & 6 \end{smallmatrix}\right], \mathbf{c}_{123} = \left[\begin{smallmatrix} 12 \\ 12 \\ -8 \end{smallmatrix}\right].

Problem 33.19

Write a system of equations defining the given plane:

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(1)

x=c28+tc33\mathbf{x} = \mathbf{c}_{28} + t\mathbf{c}_{33}. c28=[−16],c33=[21]\mathbf{c}_{28} = \left[\begin{smallmatrix} -1 \\ 6 \end{smallmatrix}\right], \mathbf{c}_{33} = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right].

(2)

x=c83+t1c84+t2c66\mathbf{x} = \mathbf{c}_{83} + t_{1} \mathbf{c}_{84} + t_{2} \mathbf{c}_{66}. c83=[123],c84=[431],c66=[111]\mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{84} = \left[\begin{smallmatrix} 4 \\ 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right].

(3)

x=c147+tc146\mathbf{x} = \mathbf{c}_{147} + t\mathbf{c}_{146}. c147=[5−2−1],c146=[141]\mathbf{c}_{147} = \left[\begin{smallmatrix} 5 \\ -2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{146} = \left[\begin{smallmatrix} 1 \\ 4 \\ 1 \end{smallmatrix}\right].

(4)

x=c168+tc207\mathbf{x} = \mathbf{c}_{168} + t\mathbf{c}_{207}. c168=[0100],c207=[1112]\mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{207} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 2 \end{smallmatrix}\right].

(5)

x=c199+t1c166+t2c200\mathbf{x} = \mathbf{c}_{199} + t_{1} \mathbf{c}_{166} + t_{2} \mathbf{c}_{200}. c199=[3−572],c166=[1111],c200=[−18−65]\mathbf{c}_{199} = \left[\begin{smallmatrix} 3 \\ -5 \\ 7 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{200} = \left[\begin{smallmatrix} -1 \\ 8 \\ -6 \\ 5 \end{smallmatrix}\right].

Problem 33.20

Write the equation of the hyperplane in a four-dimensional space passing through the point M(−1,2,3,5)M(- 1, 2, 3, 5) parallel to the hyperplane 2x1+3x2−4x3+x4+5=02x_{1} + 3x_{2} - 4x_{3} + x_{4} + 5 = 0.

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Problem 33.21

Write the equation of the line in a four-dimensional space passing through the point M(−1,3,4,0)M(- 1, 3, 4, 0) parallel to the line x1=2+3t,x2=−1+t,x3=7t,x4=2−tx_{1} = 2 + 3t, x_{2} = - 1 + t, x_{3} = 7t, x_{4} = 2 - t.

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Problem 33.22

Write the equations of the three-dimensional plane in a five-dimensional space passing:

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(1)

through the point M(0,1,−1,3,4)M(0, 1, - 1, 3, 4) parallel to the three-dimensional plane x1+2x2+3x3=x4,x1+x2+x3+x4=2x5x_{1} + 2x_{2} + 3x_{3} = x_{4}, x_{1} + x_{2} + x_{3} + x_{4} = 2x_{5}

(2)

through the points M1(1,3,1,0,1)M_{1}(1, 3, 1, 0, 1) and M2(0,0,1,1,−1)M_{2}(0, 0, 1, 1, - 1) parallel to the two-dimensional plane x1+x2−1=0,x1−x3+x4=0x_{1} + x_{2} - 1 = 0, x_{1} - x_{3} + x_{4} = 0, x1+x3−x5+1=0x_{1} + x_{3} - x_{5} + 1 = 0

(3)

through the points M1(−1,2,0,0,4),M2(1,1,1,1,1),M3(0,1M_{1}(- 1, 2, 0, 0, 4), M_{2}(1, 1, 1, 1, 1), M_{3}(0, 1, 3,−1,1)3, - 1, 1) parallel to the line x1=1+2t,x2=3−t,x3=4,x4=1+t,x5=−tx_{1} = 1 + 2t, x_{2} = 3 - t, x_{3} = 4, x_{4} = 1 + t, x_{5} = - t.

Problem 33.23

Let l\mathrm{l} and m\mathrm{m} be two planes in an affine space with directing subspaces L\mathcal{L} and M\mathcal{M} respectively, passing: l\mathrm{l} — through a point AA,  m\mathrm{~ m} — through a point BB. Prove that:

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the intersection l∩m\mathrm{l}\cap \mathrm{m} is nonempty if and only if the vector AB→\overrightarrow {AB} belongs to the subspace L+M\mathcal{L} + \mathcal{M}

(2)

if the planes l\mathrm{l} and m\mathrm{m} intersect, then the intersection l∩m\mathrm{l}\cap \mathrm{m} is a plane with directing subspace L∩M\mathcal{L} \cap \mathcal{M}.

Problem 33.24

Let two planes of dimensions k1k_{1} and k2k_{2} in an nn-dimensional affine space have a common point, and k1+k2>nk_{1} + k_{2} > n. Prove that the dimension of the intersection of the given planes is not less than k1+k2−nk_{1} + k_{2} - n. Give formulations of this statement for all possible cases for n=3n = 3 and n=4n = 4.

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Problem 33.25

Let a plane l\mathrm{l} with directing subspace L\mathcal{L} pass through a point AA, and a plane m\mathrm{m} with directing subspace M\mathcal{M} pass through a point BB not coinciding with AA. Prove that there exists a unique plane of smallest dimension containing both l\mathrm{l} and m\mathrm{m}; moreover, the directing subspace of the required plane is the sum L+M+P\mathcal{L} + \mathcal{M} + \mathcal{P}, where P\mathcal{P} is the subspace spanned by the vector AB→\overrightarrow {AB}.

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Problem 33.26

State and prove a statement analogous to the statement in problem 33. 25, for three planes.

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Problem 33.27

Write the equations of the given plane in a four-dimensional space:

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(1)

of the two-dimensional plane containing the point A(−1,0,2,3)A(- 1, 0, 2, 3) and the line x1=1−t,x2=3+2t,x3=1+t,x4=3tx_{1} = 1 - t, x_{2} = 3 + 2t, x_{3} = 1 + t, x_{4} = 3t

(2)

of the two-dimensional plane containing the parallel lines x1=−1+2t,x2=t,x3=0,x4=−5−tx_{1} = - 1 + 2t, x_{2} = t, x_{3} = 0, x_{4} = - 5 - t and x1=3+2tx_{1} = 3 + 2t, x2=−4+t,x3=1,x4=−tx_{2} = - 4 + t, x_{3} = 1, x_{4} = - t

(3)

of the three-dimensional plane containing the point A(−3,0,1,0)A(- 3, 0, 1, 0) and the two-dimensional plane x1−x2+x3−1=0,x1+x2+x4=0x_{1} - x_{2} + x_{3} - 1 = 0, x_{1} + x_{2} + x_{4} = 0.

Problem 33.28

Write the equations of the plane of smallest dimension containing two given planes of a five-dimensional space:

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(1)

the lines x1=1−t,x2=2+3t,x3=4t,x4=−t,x5=3x_{1} = 1 - t, x_{2} = 2 + 3t, x_{3} = 4t, x_{4} = - t, x_{5} = 3 and x1=2+t,x2=2t,x3=1+t,x4=−1+2t,x5=3−tx_{1} = 2 + t, x_{2} = 2t, x_{3} = 1 + t, x_{4} = - 1 + 2t, x_{5} = 3 - t

(2)

the line x1=2+t,x2=−t,x3=−1+t,x4=1+2tx_{1} = 2 + t, x_{2} = - t, x_{3} = - 1 + t, x_{4} = 1 + 2t, x5=−3tx_{5} = - 3t and the two-dimensional plane x1=t1+3t2,x2=−1+4t1−−t2,x3=−3+t1+t2,x4=4−t1+t2,x5=−2+t2x_{1} = t_{1} + 3t_{2}, x_{2} = - 1 + 4t_{1} - - t_{2}, x_{3} = - 3 + t_{1} + t_{2}, x_{4} = 4 - t_{1} + t_{2}, x_{5} = - 2 + t_{2}

(3)

the two-dimensional planes x1−x3+x4−1=0,x1+2x4−−x5−2=0,x2+x3−2=0x_{1} - x_{3} + x_{4} - 1 = 0, x_{1} + 2x_{4} - - x_{5} - 2 = 0, x_{2} + x_{3} - 2 = 0 and x1=x2=x3=1x_{1} = x_{2} = x_{3} = 1.

Problem 33.29
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Prove that if two planes in an nn-dimensional space are absolutely skew, then the sum of their dimensions does not exceed n−1n - 1.

(2)

Prove that if two planes in an nn-dimensional space are skew parallel to an rr-dimensional plane, then the sum of their dimensions does not exceed n+r−1n + r - 1.

Problem 33.30

Investigate the relative position of a line and a two-dimensional plane in a four-dimensional space, if the two-dimensional plane is given by the equations x1−2x3+1=0,x1+2x2−3x3+x4−2=0x_{1} - 2x_{3} + 1 = 0, x_{1} + 2x_{2} - 3x_{3} + x_{4} - 2 = 0, and the line is given parametrically:

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(1)

x1=3+2t,x2=5,x3=2+t,x4=tx_{1} = 3 + 2t, \quad x_{2} = 5, \quad x_{3} = 2 + t, \quad x_{4} = t

(2)

x1=−2+3t,x2=3−t,x3=−1+2t,x4=−4+4tx_{1} = - 2 + 3t, \quad x_{2} = 3 - t, \quad x_{3} = - 1 + 2t, \quad x_{4} = - 4 + 4t

(3)

x1=6+t,x2=5−t,x3=1+2t,x4=1+3tx_{1} = 6 + t, \quad x_{2} = 5 - t, \quad x_{3} = 1 + 2t, \quad x_{4} = 1 + 3t

(4)

x1=−1+2t,x2=1+t,x3=t,x4=1−tx_{1} = - 1 + 2t, \quad x_{2} = 1 + t, \quad x_{3} = t, \quad x_{4} = 1 - t.

Problem 33.31

Investigate the relative position of two two-dimensional planes in a five-dimensional space, if the first plane is given by the equations x1=x2=1,x3+x4=x5x_{1} = x_{2} = 1, x_{3} + x_{4} = x_{5}, and the second by parametric equations:

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(1)

x1=2+t1,x2=3,x3=3+2t2,x4=4,x5=5+t1+t2x_{1} = 2 + t_{1}, \quad x_{2} = 3, \quad x_{3} = 3 + 2t_{2}, \quad x_{4} = 4, \quad x_{5} = 5 + t_{1} + t_{2}

(2)

x1=−t1,x2=3+2t1,x3=2+t1,x4=1+t1−t2x_{1} = - t_{1}, \quad x_{2} = 3 + 2t_{1}, \quad x_{3} = 2 + t_{1}, \quad x_{4} = 1 + t_{1} - t_{2}, x5=2+t2x_{5} = 2 + t_{2}

(3)

x1=2+t1+t2,x2=3+t1+t2,x3=3+2t1+t2x_{1} = 2 + t_{1} + t_{2}, \quad x_{2} = 3 + t_{1} + t_{2}, \quad x_{3} = 3 + 2t_{1} + t_{2}, x4=4+t1,x5=5−2t2x_{4} = 4 + t_{1}, \quad x_{5} = 5 - 2t_{2}

(4)

x1=t1−t2,x2=1,x3=t1,x4=1−t2,x5=3−−t1+t2x_{1} = t_{1} - t_{2}, \quad x_{2} = 1, \quad x_{3} = t_{1}, \quad x_{4} = 1 - t_{2}, \quad x_{5} = 3 - - t_{1} + t_{2}

(5)

x1=1,x2=4,x3=1+t1+t2,x4=2+2t1−2t2x_{1} = 1, \quad x_{2} = 4, \quad x_{3} = 1 + t_{1} + t_{2}, \quad x_{4} = 2 + 2t_{1} - 2t_{2}, x5=−5+3t1−t2x_{5} = - 5 + 3t_{1} - t_{2}

(6)

x1=1,x2=1,x3=2+2t1+t2,x4=−3+t1−3t2x_{1} = 1, \quad x_{2} = 1, \quad x_{3} = 2 + 2t_{1} + t_{2}, \quad x_{4} = - 3 + t_{1} - 3t_{2}, x5=−1+3t1−2t2x_{5} = - 1 + 3t_{1} - 2t_{2}.

Problem 33.32

Prove that two lines in a four-dimensional space, given by the equations x=c209+tc197\mathbf{x} = \mathbf{c}_{209} + t\mathbf{c}_{197} and x=c210+tc201\mathbf{x} = \mathbf{c}_{210} + t\mathbf{c}_{201}, have a unique common point. Find the coordinates of this point and write the equations of the two-dimensional plane containing the given lines. c209=[5199],c197=[1122],c210=[15−25],c201=[1−75−2]\mathbf{c}_{209} = \left[\begin{smallmatrix} 5 \\ 1 \\ 9 \\ 9 \end{smallmatrix}\right], \mathbf{c}_{197} = \left[\begin{smallmatrix} 1 \\ 1 \\ 2 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{210} = \left[\begin{smallmatrix} 1 \\ 5 \\ -2 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{201} = \left[\begin{smallmatrix} 1 \\ -7 \\ 5 \\ -2 \end{smallmatrix}\right].

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Problem 33.33

The points of an affine space are polynomials of degree at most 4, and the vectors are the same polynomials: p1(t)p2(t)‾=p2(t)−p1(t)\overline{p_{1}(t) p_{2}(t)} = p_{2}(t) - p_{1}(t). The first line contains the points 2t4−2t2t^{4} - 2t and t4+t3−tt^{4} + t^{3} - t, the second line contains the points 5+10t2+2t35 + 10t^{2} + 2t^{3} and −1−2t2+2t3- 1 - 2t^{2} + 2t^{3}. Prove that these lines have a unique common point, and find this point (polynomial).

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Problem 33.34

Write the parametric equations of the line in a four-dimensional space containing the point with coordinate column c211\mathbf{c}_{211} and intersecting the lines x=c212+tc202\mathbf{x} = \mathbf{c}_{212} + t\mathbf{c}_{202} and x=c213+tc210\mathbf{x} = \mathbf{c}_{213} + t\mathbf{c}_{210}; find the coordinates of the points of intersection. c211=[14−15],c212=[3148],c202=[1013],c213=[417−518],c210=[15−25]\mathbf{c}_{211} = \left[\begin{smallmatrix} 1 \\ 4 \\ -1 \\ 5 \end{smallmatrix}\right], \mathbf{c}_{212} = \left[\begin{smallmatrix} 3 \\ 1 \\ 4 \\ 8 \end{smallmatrix}\right], \mathbf{c}_{202} = \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{213} = \left[\begin{smallmatrix} 4 \\ 17 \\ -5 \\ 18 \end{smallmatrix}\right], \mathbf{c}_{210} = \left[\begin{smallmatrix} 1 \\ 5 \\ -2 \\ 5 \end{smallmatrix}\right].

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Problem 33.35

The system of points A1,…,Ak,B1,…,BjA_{1}, \ldots , A_{k}, B_{1}, \ldots , B_{j} is independent. Prove that there exist disjoint planes l\mathrm{l} and m\mathrm{m} of dimensions k−1k - 1 and j−1j - 1 respectively such that the plane l\mathrm{l} contains the points A1,…,AkA_{1}, \ldots , A_{k}, and the plane m\mathrm{m} contains the points B1,…,BjB_{1}, \ldots , B_{j}.

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Problem 33.36

Let l\mathrm{l} and m\mathrm{m} be planes of an affine space such that the space of vectors of the affine space is the direct sum of the directing subspaces L\mathcal{L} and M\mathcal{M} of these planes. Prove that:

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(1)

the projection of any point of the affine space onto the plane l\mathrm{l} parallel to the plane m\mathrm{m} is uniquely defined

(2)

the projection of any vector AB→\overrightarrow {AB} onto the plane l\mathrm{l} parallel to the plane m\mathrm{m} is the projection of this vector onto the subspace L\mathcal{L} parallel to the subspace M\mathcal{M}.

Problem 33.37

Find the coordinates of the projection of the point M(5,0,−3,4)M(5, 0, - 3, 4) of a four-dimensional space:

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(1)

onto the hyperplane x1+x2−x3+2x4=2x_{1} + x_{2} - x_{3} + 2x_{4} = 2 parallel to the line x1=1−t,x2=3+4t,x3=3t,x4=1+tx_{1} = 1 - t, x_{2} = 3 + 4t, x_{3} = 3t, x_{4} = 1 + t

(2)

onto the two-dimensional plane x1−x2+x3+1=0,x1+x2=x4x_{1} - x_{2} + x_{3} + 1 = 0, x_{1} + x_{2} = x_{4} parallel to the two-dimensional plane x1+x2+x3+x4=0,x1−−2x4−3=0x_{1} + x_{2} + x_{3} + x_{4} = 0, x_{1} - - 2x_{4} - 3 = 0.

Problem 33.38

Is the set of points in an nn-dimensional affine space (n=1,2,…)(n = 1, 2, \ldots ), whose coordinates x1,…,xnx_{1}, \ldots , x_{n} in a Cartesian coordinate system satisfy the condition, convex:

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(1)

a1x1+…+anxn+a0=0a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} = 0

(2)

a1x1+…+anxn+a0⩾0a_{1} x_{1} + \ldots + a_{n} x_{n} + a_{0} \geqslant 0

(3)

λ1x12+…+λnxn2⩽1\lambda_{1} x_{1}^{2} + \ldots + \lambda_{n} x_{n}^{2} \leqslant 1, where λi⩾0,i=1,…,n\lambda_{i} \geqslant 0, \quad i = 1, \ldots , n

(4)

λ1x12+…+λnxn2⩾1\lambda_{1} x_{1}^{2} + \ldots + \lambda_{n} x_{n}^{2} \geqslant 1, where λi⩾0,i=1,…,n\lambda_{i} \geqslant 0, \quad i = 1, \ldots , n

(5)

x1⩾0,x2⩾0,…,xn⩾0x_{1} \geqslant 0, x_{2} \geqslant 0, \ldots , x_{n} \geqslant 0;

(6)

x1x2…xn⩾0x_{1} x_{2} \ldots x_{n} \geqslant 0?

Problem 33.39

Prove the convexity of a kk-dimensional parallelepiped.

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Problem 33.40

Prove that the intersection of convex sets is a convex set.

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Problem 33.41

Find the projection of the four-dimensional simplex bounded by the hyperplanes x1=0,x2=0,x3=0,x4=0x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0 and x1+x2+x3+x4=1x_{1} + x_{2} + x_{3} + x_{4} = 1, onto the hyperplane x1+x2+x3+x4=0x_{1} + x_{2} + x_{3} + x_{4} = 0 parallel to the line x1=x2=x3=x4x_{1} = x_{2} = x_{3} = x_{4}.

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Problem 33.42

Prove that all diagonals of a parallelepiped intersect at one point, called the center of the parallelepiped.

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Problem 33.43

For a kk-dimensional parallelepiped, find the number of:

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(1)

distinct pp-dimensional faces

(2)

distinct diagonals.

Problem 33.44

Determine the shape and vertices of the cross-sections of the four-dimensional parallelepiped −1⩽xi⩽1,i=1,2,3,4- 1 \leqslant x_{i} \leqslant 1, i = 1, 2, 3, 4, by the hyperplane x1+x2+x3+x4=0x_{1} + x_{2} + x_{3} + x_{4} = 0.

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Section 34Point Euclidean spaces[37/73%]
Problem 34.1

Verify that the distance ρ(A,B)\rho (A, B) between points AA and BB in a point Euclidean space has the following properties:

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(1)

ρ(A,B)=ρ(B,A)\rho (A, B) = \rho (B, A) for any points AA and BB

(2)

ρ(A,B)⩽ρ(A,C)+ρ(B,C)\rho (A, B) \leqslant \rho (A, C) + \rho (B, C) for any points A,B,CA, B, C

(3)

for any point CC such that AC→=λAB→\overrightarrow {AC} = \lambda \overrightarrow {AB}, the equality ρ(A,C)=∣λ∣ρ(A,B)\rho (A, C) = \left|\lambda \right| \rho (A, B) holds.

Problem 34.2

Find the lengths of the sides and the interior angles of the triangle ABCABC given by the coordinates of its vertices:

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(1)

A(−1,0,−1,2),B(0,2,0,3),C(2,1,1,2)A(- 1, 0, - 1, 2), \quad B(0, 2, 0, 3), \quad C(2, 1, 1, 2)

(2)

A(1,2,2,−1),B(3,0,3,−1),C(2,1,1,0)A(1, 2, 2, - 1), B(3, 0, 3, - 1), \quad C(2, 1, 1, 0)

(3)

A(0,1,−1,2,−1),B(4,1,1,2,3),C(3,4,2,5,−1)A(0, 1, - 1, 2, - 1), \quad B(4, 1, 1, 2, 3), C(3, 4, 2, 5, - 1).

Problem 34.3

Prove that the set of points equidistant from two given distinct points AA and BB is the hyperplane passing through the midpoint of the segment ABAB perpendicular to this segment.

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Problem 34.4

Find the center and radius of the sphere circumscribed about the four-dimensional simplex given by the coordinates of its vertices:

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(1)

A0(4,−2,−1,−1),A1(1,1,2,2),A2(3,1,0,0)A_{0}(4, - 2, - 1, - 1), A_{1}(1, 1, 2, 2), A_{2}(3, 1, 0, 0), A3(0,2,3,−1),A4(1,−5,−4,2)A_{3}(0, 2, 3, - 1), \quad A_{4}(1, - 5, - 4, 2)

(2)

A0(3,3,1,−1),A1(1,3,3,1),A2(0,3,4,−1)A_{0}(3, 3, 1, - 1), A_{1}(1, 3, 3, 1), A_{2}(0, 3, 4, - 1), A3(2,1,2,3),A4(2,3,0,1)A_{3}(2, 1, 2, 3), \quad A_{4}(2, 3, 0, 1).

Problem 34.5

A hyperplane m\mathrm{m} in a four-dimensional point Euclidean space contains a tetrahedron given by the coordinates of its vertices: A1(4,4,−1,1),A2(−2,−8,−5,1)A_{1}(4, 4, - 1, 1), A_{2}(- 2, - 8, - 5, 1), A3(3,3,1,3),A4(1,−2,4,1)A_{3}(3, 3, 1, 3), A_{4}(1, - 2, 4, 1). The hyperplane m is regarded as a three-dimensional point Euclidean space. Find, in this space, the center and radius of the sphere circumscribed about the given tetrahedron.

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Problem 34.6

Prove that the distance from a point AA to a kk-dimensional plane m\mathrm{m} equals:

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(1)

the distance between the point AA and its orthogonal projection onto m\mathrm{m}

(2)

the length of the orthogonal component of the vector AB→(B\overrightarrow {AB}(B is an arbitrary point of m\mathrm{m}) with respect to the directing subspace of the plane m\mathrm{m}.

Problem 34.7

Let l\mathrm{l} and m\mathrm{m} be planes with directing subspaces L\mathcal{L} and M\mathcal{M} respectively, passing: l\mathrm{l} — through a point AA and m\mathrm{m} — through a point BB. Prove that the distance between the planes l\mathrm{l} and m\mathrm{m} equals the length of the orthogonal component of the vector AB→\overrightarrow {AB} with respect to the subspace L+M\mathcal{L} + \mathcal{M}.

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Problem 34.8

A hyperplane m\mathrm{m} is given by the equation a1x1+……+anxn+a0=0a_{1} x_{1} + \ldots \ldots + a_{n} x_{n} + a_{0} = 0. Prove that:

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(1)

the vector with coordinate column (a1,…,an)T\left(a_{1}, \ldots , a_{n}\right)^{T} is orthogonal to m

(2)

the distance from the point A(y1,…,yn)A\left(y_{1}, \ldots , y_{n}\right) to m equals

∣a1y1+…+anyn+a0∣/a12+…+an2. \left|a_{1} y_{1} + \ldots + a_{n} y_{n} + a_{0}\right| / \sqrt{a_{1}^{2} + \ldots + a_{n}^{2}}.
Problem 34.9

A point AA is given by coordinates, and a hyperplane m\mathrm{m} by an equation. Find the distance from AA to m\mathrm{m}, if:

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(1)

A(9,2,−3,1),m:3x1+x2−x3−5x4+3=0A(9, 2, - 3, 1), \quad \mathrm{m}: 3x_{1} + x_{2} - x_{3} - 5x_{4} + 3 = 0

(2)

A(1,−3,0,−2,4),m:2x1−5x2+x3+3x4+5x5−7=0A(1, - 3, 0, - 2, 4), \mathrm{m}: 2x_{1} - 5x_{2} + x_{3} + 3x_{4} + 5x_{5} - 7 = 0.

Problem 34.10

Write the equation of a hyperplane parallel to a hyperplane m\mathrm{m} and located at a given distance dd from m\mathrm{m}, if:

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(1)

m:5x1+2x2−4x3+2x4=3,d=2\mathrm{m}: 5x_{1} + 2x_{2} - 4x_{3} + 2x_{4} = 3, \quad d = 2

(2)

m:x1−4x2+2x3+2x4=4,d=5\mathrm{m}: x_{1} - 4x_{2} + 2x_{3} + 2x_{4} = 4, \quad d = 5

(3)

m:2x1−x2−x3+x4+3x5=−5,d=3\mathrm{m}: 2x_{1} - x_{2} - x_{3} + x_{4} + 3x_{5} = - 5, \quad d = 3.

Problem 34.11

Find the orthogonal projection of a point AA onto a hyperplane m\mathrm{m}:

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(1)

A(7,−1,6,1),m:3x1−x2+2x3+x4=5A(7, - 1, 6, 1), \quad \mathrm{m}: 3x_{1} - x_{2} + 2x_{3} + x_{4} = 5

(2)

A(1,2,8,−2)A(1, 2, 8, - 2), m: 2x1−2x3+x4=112x_{1} - 2x_{3} + x_{4} = 11

(3)

A(3,0,−1,2,6)A(3, 0, - 1, 2, 6), m: 5x1+3x2−2x3−x4+4x5=−165x_{1} + 3x_{2} - 2x_{3} - x_{4} + 4x_{5} = - 16.

Problem 34.12

Points AA and BB are given by their coordinates. Find the orthogonal projection of the vector AB‾\overline{AB} onto a hyperplane m\mathrm{m}, if:

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(1)

A(−3,0,1,3),B(5,2,2,3),m:2x1+x2−x4=3A(- 3, 0, 1, 3), \quad B(5, 2, 2, 3), \quad \mathrm{m}: 2x_{1} + x_{2} - x_{4} = 3

(2)

A(3,3,−8,−3,4),B(3,2,−1,−2,2),m:x1+x2−−2x3+x4+x5=5A(3, 3, - 8, - 3, 4), \quad B(3, 2, - 1, - 2, 2), \mathrm{m}: \quad x_{1} + x_{2} - - 2x_{3} + x_{4} + x_{5} = 5.

Problem 34.13

Find the ratio of the length of the orthogonal projection of an arbitrary edge of an nn-dimensional cube onto its diagonal to the length of the diagonal.

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Problem 34.14

Find the point orthogonally symmetric to a point AA with respect to a hyperplane m\mathrm{m}:

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(1)

A(5,5,3,3),m:2x1+3x2+x3+2x4+2=0A(5, 5, 3, 3), \quad \mathrm{m}: 2x_{1} + 3x_{2} + x_{3} + 2x_{4} + 2 = 0

(2)

A(3,5,−3,5),m:x1−3x2+4x3−5x4−2=0A(3, 5, - 3, 5), \mathrm{m}: x_{1} - 3x_{2} + 4x_{3} - 5x_{4} - 2 = 0

(3)

A(3,6,3,8,1),m:x1−x2−2x4+2x5−3=0A(3, 6, 3, 8, 1), \mathrm{m}: x_{1} - x_{2} - 2x_{4} + 2x_{5} - 3 = 0.

Problem 34.15

Find the orthogonal projection of a point AA onto a line:

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(1)

A(1,−5,2,0)A(1, - 5, 2, 0); l: x1=4+t,x2=3+2t,x3=−3−tx_{1} = 4 + t, \quad x_{2} = 3 + 2t, \quad x_{3} = - 3 - t, x4=7+3tx_{4} = 7 + 3t

(2)

A(−2,1,4,2)A(- 2, 1, 4, 2); l: x1=−3+2t,x2=3−t,x3=−1+tx_{1} = - 3 + 2t, \quad x_{2} = 3 - t, \quad x_{3} = - 1 + t, x4=−3+tx_{4} = - 3 + t

(3)

A(2,4,3,−1,1)A(2, 4, 3, - 1, 1); l: x1=2−2t,x2=−1+3t,x3=−1+2t,x4=2+t,x5=−tx_{1} = 2 - 2t, \quad x_{2} = - 1 + 3t, \quad x_{3} = - 1 + 2t, \quad x_{4} = 2 + t, \quad x_{5} = - t.

Problem 34.16

A point AA does not belong to a plane m\mathrm{m}. Prove that there exists a unique line passing through the point AA, intersecting m\mathrm{m} and perpendicular to m\mathrm{m}.

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Problem 34.17

Write the equations of the perpendicular dropped from a point AA onto a line l\mathrm{l}:

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(1)

A(1,−3,−2,4)A(1, - 3, - 2, 4); l: x1=4+3t,x2=2+t,x3=3+tx_{1} = 4 + 3t, \quad x_{2} = 2 + t, \quad x_{3} = 3 + t, x4=−1−tx_{4} = - 1 - t

(2)

A(1,−3,−1,3)A(1, - 3, - 1, 3); l: x1=2+t,x2=1−2t,x3=−1+2t,x4=tx_{1} = 2 + t, \quad x_{2} = 1 - 2t, \quad x_{3} = - 1 + 2t, \quad x_{4} = t

(3)

A(4,0,1,1,1)A(4, 0, 1, 1, 1); l: x1=t,x2=3−2t,x3=−2+tx_{1} = t, \quad x_{2} = 3 - 2t, \quad x_{3} = - 2 + t, x4=−3+2t,x5=tx_{4} = - 3 + 2t, \quad x_{5} = t.

Problem 34.18

Find the point orthogonally symmetric to a point AA with respect to a line l\mathrm{l}:

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(1)

A(4,1,−1,−1)A(4, 1, - 1, - 1), l\mathrm{l} is the line of problem 34.15,1)34. 15, 1)

(2)

A(2,5,−3,−2)A(2, 5, - 3, - 2), l\mathrm{l} is the line of problem 34.15,2)34. 15, 2).

Problem 34.19

Find the angle between a vector given by the coordinate column a, and a hyperplane m\mathrm{m}, if:

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(1)

a=[0101], m:3x1−x2+x3−5x4=2\mathbf{a} = \left[\begin{smallmatrix} 0\\ 1\\ 0\\ 1\end{smallmatrix}\right], \quad \mathrm{~ m}: 3x_{1} - x_{2} + x_{3} - 5x_{4} = 2

(2)

a=[1−111], m:3x1−x2+2x3+2x4=5\mathbf{a} = \left[\begin{smallmatrix} 1\\ - 1\\ 1\\ 1\end{smallmatrix}\right], \quad \mathrm{~ m}: 3x_{1} - x_{2} + 2x_{3} + 2x_{4} = 5

(3)

a=[1−32−1−1], m:x1+x2−2x3+3x4−x5=1\mathbf{a} = \left[\begin{smallmatrix} 1\\ - 3\\ 2\\ - 1\\ - 1\end{smallmatrix}\right], \quad \mathrm{~ m}: x_{1} + x_{2} - 2x_{3} + 3x_{4} - x_{5} = 1.

Problem 34.20

Find the angle between the lines l1l_{1} and l2l_{2}, if:

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(1)

l1:x1=4+t,x2=−2t,x3=1−t,x4=2;l2:x1=3l_{1}: x_{1} = 4 + t, x_{2} = - 2t, x_{3} = 1 - t, x_{4} = 2; l_{2}: x_{1} = 3, x2=t,x3=5+t,x4=−1x_{2} = t, \quad x_{3} = 5 + t, \quad x_{4} = - 1

(2)

l1:x1=1+t,x2=2+t,x3=3+t,x4=2t,x5=1−t;l2:x1=t,x2=5,x3=−1+t,x4=3−2t,x5=2t\mathrm{l}_{1}: x_{1} = 1 + t, \quad x_{2} = 2 + t, \quad x_{3} = 3 + t, \quad x_{4} = 2t, \quad x_{5} = 1 - t; \quad \mathrm{l}_{2}: x_{1} = t, \quad x_{2} = 5, \quad x_{3} = - 1 + t, \quad x_{4} = 3 - 2t, \quad x_{5} = 2t.

Problem 34.21

Find the distance from a point AA to a line l\mathrm{l}:

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(1)

A(0,3,2,−5)A(0, 3, 2, - 5); l: x1=1+t,x2=−t,x3=2+2tx_{1} = 1 + t, \quad x_{2} = - t, x_{3} = 2 + 2t, x4=−2+2tx_{4} = - 2 + 2t

(2)

A(2,−2,1,5);l:x1=3+t,x2=−1+t,x3=2+tA(2, - 2, 1, 5); \quad \mathrm{l}: x_{1} = 3 + t, \quad x_{2} = - 1 + t, \quad x_{3} = 2 + t, x4=−tx_{4} = - t

(3)

A(3,3,1,0,0);l:x1=2+3t,x2=1+2t,x3=−tA(3, 3, 1, 0, 0); \quad \mathrm{l}: \quad x_{1} = 2 + 3t, \quad x_{2} = 1 + 2t, \quad x_{3} = - t, x4=1+t,x5=−1−tx_{4} = 1 + t, \quad x_{5} = - 1 - t

(4)

A(1,−1,−1,1)A(1, - 1, - 1, 1); l: x1+x2+2x3+1=0,3x2+2x3−−x4−1=0,x1−x2+x3+x4+2=0x_{1} + x_{2} + 2x_{3} + 1 = 0, \quad 3x_{2} + 2x_{3} - - x_{4} - 1 = 0, x_{1} - x_{2} + x_{3} + x_{4} + 2 = 0.

Problem 34.22

A line l1l_{1} with directing vector a1a_{1} passes through a point A1A_{1}, and a line l2\mathrm{l}_{2} with directing vector a2a_{2} passes through a point A2A_{2}. Prove that:

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(1)

if a1a_{1} and a2a_{2} are not collinear, then the square of the distance between l1l_{1} and l2l_{2} equals det⁡Γ(A1A2→,a1,a2)/det⁡Γ(a1,a2)\operatorname {det} \Gamma \left(\overrightarrow {A_{1} A_{2}}, a_{1}, a_{2}\right) / \operatorname {det} \Gamma \left(a_{1}, a_{2}\right)

(2)

if a1a_{1} and a2a_{2} are collinear, then the square of the distance between l1l_{1} and l2l_{2} equals det⁡Γ(A1A2→,a1)/∣a1∣2\operatorname {det} \Gamma \left(\overrightarrow {A_{1} A_{2}}, a_{1}\right) / \left|a_{1}\right|^{2}.

Problem 34.23

Find the distance between the lines l1l_{1} and l2\mathrm{l}_{2}:

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(1)

l1:x1=1+t,x2=−1,x3=−t,x4=−2+t\mathrm{l}_{1}: \quad x_{1} = 1 + t, \quad x_{2} = - 1, \quad x_{3} = - t, \quad x_{4} = - 2 + t; l2:x1=4+t,x2=2t,x3=1+t,x4=t\mathrm{l}_{2}: x_{1} = 4 + t, \quad x_{2} = 2t, \quad x_{3} = 1 + t, \quad x_{4} = t

(2)

l1:x1=2+t,x2=−1−2t,x3=2+2t,x4=1−t\mathrm{l}_{1}: \quad x_{1} = 2 + t, \quad x_{2} = - 1 - 2t, \quad x_{3} = 2 + 2t, \quad x_{4} = 1 - t; l2:x1=3−t,x2=1+2t,x3=−1−2t,x4=2+tl_{2}: x_{1} = 3 - t, \quad x_{2} = 1 + 2t, \quad x_{3} = - 1 - 2t, \quad x_{4} = 2 + t

(3)

l1:x1=3+t,x2=2,x3=t,x4=3+t,x5=−t\mathrm{l}_{1}: \quad x_{1} = 3 + t, \quad x_{2} = 2, \quad x_{3} = t, \quad x_{4} = 3 + t, \quad x_{5} = - t; l2:x1=1+2t,x2=2t,x3=1−t,x4=t,x5=2l_{2}: x_{1} = 1 + 2t, \quad x_{2} = 2t, \quad x_{3} = 1 - t, \quad x_{4} = t, \quad x_{5} = 2

(4)

l1:x1=1+t,x2=2t,x3=1−t,x4=−1+t,x5=tl_{1}: x_{1} = 1 + t, \quad x_{2} = 2t, \quad x_{3} = 1 - t, \quad x_{4} = - 1 + t, \quad x_{5} = t; l2:x1=3+t,x2=−2t,x3=−1−t,x4=1+t,x5=2+tl_{2}: x_{1} = 3 + t, \quad x_{2} = - 2t, \quad x_{3} = - 1 - t, \quad x_{4} = 1 + t, \quad x_{5} = 2 + t

(5)

l1:x1=1−2t,x2=0,x3=t,x4=1+t,x5=2\mathrm{l}_{1}: \quad x_{1} = 1 - 2t, \quad x_{2} = 0, \quad x_{3} = t, \quad x_{4} = 1 + t, \quad x_{5} = 2; l2:x1=−1+t,x2=−1+t,x3=0,x4=1,x5=−2−tl_{2}: x_{1} = - 1 + t, \quad x_{2} = - 1 + t, \quad x_{3} = 0, \quad x_{4} = 1, \quad x_{5} = - 2 - t.

Problem 34.24

Write the equations of the perpendicular dropped from a point AA onto a plane m\mathrm{m}, if:

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(1)

A(3,7,−2,1);m:x1=2+t1,x2=2+t2,x3=t1+t2,x4=−t2A(3, 7, - 2, 1); \mathrm{m}: x_{1} = 2 + t_{1}, \quad x_{2} = 2 + t_{2}, \quad x_{3} = t_{1} + t_{2}, \quad x_{4} = - t_{2}

(2)

A(−3,−1,4,7,−3);m:x1=t1+t3,x2=2+t2+t3A(- 3, - 1, 4, 7, - 3); \mathrm{m}: x_{1} = t_{1} + t_{3}, \quad x_{2} = 2 + t_{2} + t_{3}, x3=2+t1,x4=1+t2+t3,x5=1+t2x_{3} = 2 + t_{1}, \quad x_{4} = 1 + t_{2} + t_{3}, \quad x_{5} = 1 + t_{2}.

Problem 34.25

Find the orthogonal projection of a point AA onto a plane m\mathrm{m}, if:

?
(1)

A(−3,2,2,−2);m:x1=2+t1+t2,x2=4+2t1A(- 3, 2, 2, - 2); \mathrm{m}: x_{1} = 2 + t_{1} + t_{2}, \quad x_{2} = 4 + 2t_{1}, x3=t1,x4=−t2x_{3} = t_{1}, \quad x_{4} = - t_{2}

(2)

A(3,2,1,4,−1)A(3, 2, 1, 4, - 1); m:x1=1+t1,x2=−1+t2,x3=2+t1+t2,x4=−2−t1,x5=t2\mathrm{m}: x_{1} = 1 + t_{1}, \quad x_{2} = - 1 + t_{2}, \quad x_{3} = 2 + t_{1} + t_{2}, \quad x_{4} = - 2 - t_{1}, \quad x_{5} = t_{2}

(3)

A(0,−1,5,1,−2)A(0, - 1, 5, 1, - 2); m: x1=1+t1,x2=t3,x3=1+t1+t2,x4=−2+t3,x5=−1+t2x_{1} = 1 + t_{1}, \quad x_{2} = t_{3}, \quad x_{3} = 1 + t_{1} + t_{2}, \quad x_{4} = - 2 + t_{3}, \quad x_{5} = - 1 + t_{2}.

Problem 34.26

Find the point orthogonally symmetric to a point AA with respect to a plane m\mathrm{m}, if:

?
(1)

A(5,3,−1,−1);m:x1=1+t1,x2=t2,x3=−2+t2,x4=−1+t1A(5, 3, - 1, - 1); \mathrm{m}: \quad x_{1} = 1 + t_{1}, \quad x_{2} = t_{2}, \quad x_{3} = - 2 + t_{2}, \quad x_{4} = - 1 + t_{1}

(2)

A(3,5,0,2,2);m:x1=t1,x2=2+t2,x3=−3+t1A(3, 5, 0, 2, 2); \mathrm{m}: x_{1} = t_{1}, \quad x_{2} = 2 + t_{2}, \quad x_{3} = - 3 + t_{1}, x4=3−t1−t2,x5=1x_{4} = 3 - t_{1} - t_{2}, \quad x_{5} = 1.

Problem 34.27

Let m\mathrm{m} be a plane with directing subspace M\mathcal{M}, passing through a point A0A_{0}, and let f1,…,fkf_{1}, \ldots , f_{k} be a basis in M\mathcal{M}. Prove that the square of the distance from a point A1A_{1} to the plane m\mathrm{m} equals

det⁡Γ(A0A1→,f1,…,fk)/det⁡Γ(f1,…,fk) \operatorname {det} \Gamma \left(\overrightarrow {A_{0} A_{1}}, f_{1}, \ldots , f_{k}\right) / \operatorname {det} \Gamma \left(f_{1}, \ldots , f_{k}\right)
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Problem 34.28

Find the distance from a point AA to a plane m:

?
(1)

in problem 34.24,1)34. 24, 1)

(2)

in problem 34.24,2)34. 24, 2).

Problem 34.29

Find the distance from a point AA to a plane m\mathrm{m} given parametrically, if:

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(1)

A(1,2,1,1);m:x1=−2t1+4t2,x2=−1+t1−t2A(1, 2, 1, 1); \mathrm{m}: \quad x_{1} = - 2t_{1} + 4t_{2}, \quad x_{2} = - 1 + t_{1} - t_{2}, x3=−t3,x4=t1−t2x_{3} = - t_{3}, \quad x_{4} = t_{1} - t_{2}

(2)

A(3,1,1,0)A(3, 1, 1, 0); m:x1=−2+t1,x2=−t1+2t2,x3=t1−t2,x4=1−t1−t2\mathrm{m}: x_{1} = - 2 + t_{1}, \quad x_{2} = - t_{1} + 2t_{2}, \quad x_{3} = t_{1} - t_{2}, \quad x_{4} = 1 - t_{1} - t_{2}

(3)

A(1,2,1,3,0)A(1, 2, 1, 3, 0); m:x1=1+t1,x2=−t1+t2,x3=1+t2,x4=−1−t2,x5=t1\mathrm{m}: \quad x_{1} = 1 + t_{1}, \quad x_{2} = - t_{1} + t_{2}, \quad x_{3} = 1 + t_{2}, \quad x_{4} = - 1 - t_{2}, \quad x_{5} = t_{1}.

Problem 34.30

Find the distance from a point AA to a plane m\mathrm{m} given by a system of linear equations, if:

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(1)

A(1,0,0,1)A(1, 0, 0, 1); m: x1+2x2+2x3−3x4=7,x1−2x3+2x4=−6x_{1} + 2x_{2} + 2x_{3} - 3x_{4} = 7, \quad x_{1} - 2x_{3} + 2x_{4} = - 6

(2)

A(1,2,0,0)A(1, 2, 0, 0); m: x1+x2−x3−x4=1,2x2−3x3+x4=2,2x1+x3−3x4=0x_{1} + x_{2} - x_{3} - x_{4} = 1, \quad 2x_{2} - 3x_{3} + x_{4} = 2, \quad 2x_{1} + x_{3} - 3x_{4} = 0.

Problem 34.31

Points AA and BB are given by their coordinates. Find the angle between the vector AB‾\overline{AB} and a plane m\mathrm{m}, if:

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(1)

A(1,2,2,3),B(4,0,0,2);m:x1=1+t1,x2=2+t2,x3=−t1+t2,x4=3A(1, 2, 2, 3), \quad B(4, 0, 0, 2); \mathrm{m}: x_{1} = 1 + t_{1}, \quad x_{2} = 2 + t_{2}, \quad x_{3} = - t_{1} + t_{2}, \quad x_{4} = 3

(2)

A(0,1,−1,0,1),B(3,1,0,1,2);m:x1=t1+t2,x2=5A(0, 1, - 1, 0, 1), B(3, 1, 0, 1, 2); \mathrm{m}: x_{1} = t_{1} + t_{2}, x_{2} = 5, x3=−t2,x4=−t1+t2,x5=2+t1x_{3} = - t_{2}, \quad x_{4} = - t_{1} + t_{2}, \quad x_{5} = 2 + t_{1}

(3)

A(−1,−1,1,0,1),B(2,1,1,1,0)A(- 1, - 1, 1, 0, 1), \quad B(2, 1, 1, 1, 0); m: x1=t1+t3x_{1} = t_{1} + t_{3}, x2=2+t2,x3=1−t2,x4=−t1+t3,x5=−2t3x_{2} = 2 + t_{2}, \quad x_{3} = 1 - t_{2}, \quad x_{4} = - t_{1} + t_{3}, \quad x_{5} = - 2t_{3}.

Problem 34.32

Planes l\mathrm{l} and m\mathrm{m} of an nn-dimensional point Euclidean space, with directing subspaces L\mathcal{L} and M\mathcal{M} respectively, pass: l\mathrm{l} — through a point AA,  m\mathrm{~ m} — through a point BB. Let g1,…,gkg_{1}, \ldots , g_{k} be a basis in the subspace L+M\mathcal{L} + \mathcal{M}. Prove that the square of the distance between the planes l\mathrm{l} and m\mathrm{m} equals

det⁡Γ(AB→,g1,…,gk)/det⁡Γ(g1,…,gk).  \operatorname {det} \Gamma \left(\overrightarrow {AB}, g_{1}, \ldots , g_{k}\right) / \operatorname {det} \Gamma \left(g_{1}, \ldots , g_{k}\right) \text{. }
?
Problem 34.33

Find the distance between the planes l\mathrm{l} and m\mathrm{m}:

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(1)

l: x1=2−2t,x2=4+t,x3=1+t,x4=0x_{1} = 2 - 2t, \quad x_{2} = 4 + t, \quad x_{3} = 1 + t, \quad x_{4} = 0; m:x1=1−2t1,x2=1+2t1+3t2,x3=1+t1,x4=1+2t1+2t2\mathrm{m}: x_{1} = 1 - 2t_{1}, \quad x_{2} = 1 + 2t_{1} + 3t_{2}, \quad x_{3} = 1 + t_{1}, \quad x_{4} = 1 + 2t_{1} + 2t_{2}

(2)

l: x1=3+t1+2t2,x2=−t1,x3=1+t1−t2x_{1} = 3 + t_{1} + 2t_{2}, \quad x_{2} = - t_{1}, \quad x_{3} = 1 + t_{1} - t_{2}, x4=−t1−t2;m:x1=2t1+t2,x2=1−3t1+t2x_{4} = - t_{1} - t_{2}; \quad \mathrm{m}: \quad x_{1} = 2t_{1} + t_{2}, \quad x_{2} = 1 - 3t_{1} + t_{2}, x3=−8−t2,x4=1+t1−t2x_{3} = - 8 - t_{2}, x_{4} = 1 + t_{1} - t_{2}

(3)

l: x1=2+t,x2=2t,x3=1,x4=t,x5=0x_{1} = 2 + t, \quad x_{2} = 2t, \quad x_{3} = 1, \quad x_{4} = t, \quad x_{5} = 0; m:x1=0,x2=1+t1+t2,x3=3+2t2,x4=2t1,x5=1+t1−t2\mathrm{m}: \quad x_{1} = 0, \quad x_{2} = 1 + t_{1} + t_{2}, \quad x_{3} = 3 + 2t_{2}, \quad x_{4} = 2t_{1}, \quad x_{5} = 1 + t_{1} - t_{2}

(4)

l: x1=1+t2,x2=t2,x3=t1,x4=t1,x5=2t1x_{1} = 1 + t_{2}, \quad x_{2} = t_{2}, \quad x_{3} = t_{1}, \quad x_{4} = t_{1}, \quad x_{5} = 2t_{1}; m:x1=t2+2t3,x2=2+t1+2t3,x3=x4=1+t1−t2+t3\mathrm{m}: x_{1} = t_{2} + 2t_{3}, \quad x_{2} = 2 + t_{1} + 2t_{3}, \quad x_{3} = x_{4} = 1 + t_{1} - t_{2} + t_{3}, x5=2+t1−t2+2t3x_{5} = 2 + t_{1} - t_{2} + 2t_{3}

(5)

l: 2x1−x3+3x4=0,2x1−2x2+3x3−3x4=82x_{1} - x_{3} + 3x_{4} = 0, \quad 2x_{1} - 2x_{2} + 3x_{3} - 3x_{4} = 8; m:x2=−3x3−2x4=2,x1−x3−x4=0\mathrm{m}: x_{2} = - 3x_{3} - 2x_{4} = 2, \quad x_{1} - x_{3} - x_{4} = 0

(6)

l: x1+x2−x3=1,2x1+x2−x4=4x_{1} + x_{2} - x_{3} = 1, \quad 2x_{1} + x_{2} - x_{4} = 4; m: x1+x2+x3=−1,x1+x3+x4=1,2x1−x2−x4=0x_{1} + x_{2} + x_{3} = - 1, \quad x_{1} + x_{3} + x_{4} = 1, \quad 2x_{1} - x_{2} - x_{4} = 0.

Problem 34.34

In an n-dimensional space, planes m1\mathrm{m}_{1} and m2\mathrm{m}_{2} of dimensions k1k_{1} and k2k_{2} respectively are absolutely skew. Prove that:

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(1)

there exists a unique plane of dimension n−−k1−k2n - - k_{1} - k_{2}, orthogonal to m1\mathrm{m}_{1} and to m2\mathrm{m}_{2} and intersecting each of these planes

(2)

there exists a unique line, orthogonal to m1\mathrm{m}_{1} and to m2\mathrm{m}_{2} and intersecting each of these planes.

Problem 34.35

Find the equations of the plane of maximal dimension orthogonal to given planes m1\mathrm{m}_{1} and m2\mathrm{m}_{2} and intersecting each of these planes, as well as the equations of the common perpendicular to m1\mathrm{m}_{1} and m2\mathrm{m}_{2}, if:

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(1)

m1\mathrm{m}_{1} and m2\mathrm{m}_{2} are the lines of problem 34.23,134. 23, 1)

(2)

m1\mathrm{m}_{1} and m2\mathrm{m}_{2} are the lines of problem 34.23,334. 23, 3)

(3)

m1\mathrm{m}_{1} and m2\mathrm{m}_{2} are the planes of problem 34.33,334. 33, 3).

Problem 34.36

Find the angle between the planes m1:x1=2+t1+t2,x2=x3=t1,x4=−1+t1−t2\mathrm{m}_{1}: x_{1} = 2 + t_{1} + t_{2}, x_{2} = x_{3} = t_{1}, x_{4} = - 1 + t_{1} - t_{2} and m2:x1=t1+2t2,x2=3+t2,x3=2−t1−2t2,x4=−t2\mathrm{m}_{2}: x_{1} = t_{1} + 2t_{2}, x_{2} = 3 + t_{2}, x_{3} = 2 - t_{1} - 2t_{2}, x_{4} = - t_{2}.

?
Problem 34.37

In a regular five-dimensional simplex A0A1A2A3A4A5A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}, find the angle:

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(1)

between the faces A0A1A2A_{0} A_{1} A_{2} and A0A3A4A_{0} A_{3} A_{4}

(2)

between the faces A0A1A2A_{0} A_{1} A_{2} and A0A3A4A5A_{0} A_{3} A_{4} A_{5}

(3)

between the faces A0A1A2A_{0} A_{1} A_{2} and A1A2A3A4A5A_{1} A_{2} A_{3} A_{4} A_{5}.