Math Problems Vault

Problem 379

Determine which of the following pairs of planes intersect, are parallel, or coincide. If the planes intersect, find the canonical equations of the line of intersection:

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(1)

$x+2 y-z+1 = 0, \quad x+y-2 z+1 = 0$

(2)

$x-3 y+7 z+1 = 0, \quad 3 x-9 y+21 z+3 = 0$ ;

(3)

$x-3 y+2 z+2 = 0, \quad 2 x-6 y+4 z+5 = 0$ ;

(4)

$\begin{cases} x = 1+2 u+v, \\ y = 3 u+2 v, \\ z = -1+u+v ; \end{cases} \quad \begin{cases} x = 4 u+3 v, \\ y = -2+u-4 v, \\ z = 3+u+2 v ; \end{cases}$

(5)

$\begin{cases} x = -3+u+2 v, \\ y = -v, \\ z = u ; \end{cases} \begin{cases} x = 1+3 u, \\ y = 2-u+v, \\ z = 8+u+2 v ; \end{cases}$

(6)

$\begin{cases} x = -3+u+4 v, \\ y = 3+u+v, \\ z = 2+u+2 v ; \end{cases} \quad x+2 y-3 z+2 = 0$ .

Problem 380

In each of the following cases, determine whether the line $\ell$ lies in the plane $P$ , is parallel to the plane $P$ , or intersects it; in the last case, find the point of intersection of the line and the plane:

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(1)

$\ell : \frac{x-1}{-1} = \frac{y+2}{7} = \frac{z+1}{7}$ , $P: 7 x+2 y-z+7 = 0$ ;

(2)

$\ell : \frac{x}{0} = \frac{y-2}{2} = \frac{z+1}{1}$ ,

P: 5 x-y+6 z = 0

(3)

$\ell : \frac{x}{1} = \frac{y-1}{1} = \frac{z+1}{1}$ ,

P: x-3 y+2 z+5 = 0

(4)

$\ell : x-3 y+1 = 0,2 x+y-5 = 0, \quad P: x+2 y-2 z-2 = 0$ ;

(5)

$\ell : x+2 y+z-1 = 0, x+y-2 = 0, P: 3 x+2 y-z-6 = 0$ .

Problem 381

Determine the relative position of the two lines in each of the following cases (they may: a) intersect, b) be parallel, c) be skew, d) coincide):

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(1)

$\begin{cases} x = 1+2 t, \\ y = 7+t, \\ z = 3+4 t ; \end{cases} \begin{cases} x = 6+3 t \\ y = -1-2 t \\ z = -2+t \end{cases}$

(2)

(3)

$x = t, y = -8-4 t, z = -3-3 t ; \quad \begin{cases} 2 x-y+2 z = 0, \\ x+y-z = 0 ; \end{cases}$

(4)

$x = 3+t, y = -1+2 t, z = 4 ; \quad \begin{cases} x-3 y+z = 0, \\ x+y-z+4 = 0 \end{cases}$

(5)

$\begin{cases} x+y+z-1 = 0, \\ y+4 z = 0 ; \end{cases}, \begin{cases} 2 x+3 y+6 z-6 = 0, \\ 3 x+4 y+7 z = 0. \end{cases}$

Problem 382

(1)(2)(3)

Two planes are given

A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, \quad A_{2} x+B_{2} y+C_{2} z+D_{2} = 0

Using the ranks $r$ and $R$ of the matrices

\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \end{smallmatrix}\right] \text{ and }\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} & D_{1} \\ A_{2} & B_{2} & C_{2} & D_{2} \end{smallmatrix}\right]

respectively, find the necessary and sufficient conditions for the planes to:

?

(1)

intersect;

(2)

be parallel;

(3)

coincide.

Problem 383

(1)(2)(3)

Two planes are given by their parametric equations

\begin{array}{ll} x = x_{1}+a_{1} u+a_{2} v, & x = x_{2}+a_{3} u+a_{4} v \\ y = y_{1}+b_{1} u+b_{2} v, & y = y_{2}+b_{3} u+b_{4} v \\ z = z_{1}+c_{1} u+c_{2} v ; & z = z_{2}+c_{3} u+c_{4} v \end{array}

Using the ranks $r$ and $R$ of the matrices

\left[\begin{smallmatrix} a_{1} & a_{2} & a_{3} & a_{4} \\ b_{1} & b_{2} & b_{3} & b_{4} \\ c_{1} & c_{2} & c_{3} & c_{4} \end{smallmatrix}\right] \quad \text{ and }\left[\begin{smallmatrix} a_{1} & a_{2} & a_{3} & a_{4} & x_{2}-x_{1} \\ b_{1} & b_{2} & b_{3} & b_{4} & y_{2}-y_{1} \\ c_{1} & c_{2} & c_{3} & c_{4} & z_{2}-z_{1} \end{smallmatrix}\right]

respectively, express the necessary and sufficient conditions for the planes to:

?

(1)

intersect;

(2)

be parallel;

(3)

coincide.

Problem 384

(1)(2)(3)

A plane $A x+B y+C z+D = 0$ and a line

\frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c}.

are given. Find the necessary and sufficient conditions for the line to:

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(1)

intersect the plane;

(2)

be parallel to it;

(3)

lie in the plane.

Problem 385

(1)(2)(3)(4)

Two lines are given

\frac{x-x_{1}}{a_{1}} = \frac{y-y_{1}}{b_{1}} = \frac{z-z_{1}}{c_{1}}, \quad \frac{x-x_{2}}{a_{2}} = \frac{y-y_{2}}{b_{2}} = \frac{z-z_{2}}{c_{2}}

Using the ranks $r$ and $R$ of the matrices

\left[\begin{smallmatrix} a_{1} & a_{2} \\ b_{1} & b_{2} \\ c_{1} & c_{2} \end{smallmatrix}\right] \quad \text{ and }\left[\begin{smallmatrix} a_{1} & a_{2} & x_{2}-x_{1} \\ b_{1} & b_{2} & y_{2}-y_{1} \\ c_{1} & c_{2} & z_{2}-z_{1} \end{smallmatrix}\right]

respectively, express the conditions necessary and sufficient for these lines to:

?

(1)

be skew;

(2)

intersect;

(3)

be parallel;

(4)

coincide.

Problem 386

(1)(2)(3)(4)

Two lines are given

\frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c}, \quad \begin{cases} A_{1} x+B_{1} y+C_{1} z+D_{1} = 0 \\ A_{2} x+B_{2} y+C_{2} z+D_{2} = 0 \end{cases}

Find the necessary and sufficient conditions for them to:

?

(1)

be skew;

(2)

intersect;

(3)

be parallel;

(4)

coincide.

Problem 387

Find the locus of points dividing, in one and the same ratio $\lambda$ , segments with endpoints on two given skew lines.

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Problem 388

Three planes are given:

\begin{gathered} A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, \quad A_{2} x+B_{2} y+C_{2} z+D_{2} = 0 \\ A_{3} x+B_{3} y+C_{3} z+D_{3} = 0 \end{gathered}

Let $r$ and $R$ be the ranks of the matrices

\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{smallmatrix}\right] \text{ and }\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} & D_{1} \\ A_{2} & B_{2} & C_{2} & D_{2} \\ A_{3} & B_{3} & C_{3} & D_{3} \end{smallmatrix}\right]

respectively; $r_{i}, R_{i}$ the ranks of the matrices obtained from the given ones by deleting the $i$ -th row, $i = 1,2,3$ . Express, in terms of $r, R, r_{i}, R_{i}$ , the necessary and sufficient conditions for:

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(1)

the three planes to have a unique common point;

(2)

the three planes to be pairwise distinct and to have a unique common line;

(3)

the three planes to intersect pairwise, with the line of intersection of each two planes parallel to the third plane (i.e., for the planes to form a prism);

(4)

two planes to be parallel, and the third plane to intersect them;

(5)

the three planes to be pairwise parallel;

(6)

two planes to coincide, and the third to intersect them;

(7)

two planes to coincide, and the third plane to be parallel to them;

(8)

the three planes to coincide.

Problem 389

Determine the relative position of the three planes in each of the following cases:

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(1)

$x+2 y-z+7 = 0, \quad x+y-2 z+4 = 0, \quad 5 x+3 z+2 = 0$ ;

(2)

$5 x-3 y+7 z+10 = 0, x-y+6 z+1 = 0, \quad 10 x-6 y+14 z+9 = 0$ ;

(3)

$x-3 y+2 z+2 = 0, \quad 2 x-6 y+4 z+3 = 0, \quad 5 x-15 y+10 z+16 = 0$ ;

(4)

$x+2 y-z+7 = 0, \quad x+y-2 z+4 = 0, \quad x-y-4 z-2 = 0$ ;

(5)

$5 x-2 y+3 = 0, \quad 3 x+z-6 = 0, \quad 8 x-2 y+z+6 = 0$ ;

(6)

$x-y+2 z+3 = 0, \quad 6 x-3 y+z-4 = 0,2 x-2 y+4 z+6 = 0$ .

Problem 390

Under what necessary and sufficient condition do the four planes

\begin{array}{ll} A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, & A_{2} x+B_{2} y+C_{2} z+D_{2} = 0 \\ A_{3} x+B_{3} y+C_{3} z+D_{3} = 0, & A_{4} x+B_{4} y+C_{4} z+D_{4} = 0 \end{array}

form a tetrahedron?

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Problem 391

(1)(2)(3)

Four planes are given

\begin{array}{ll} A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, & A_{2} x+B_{2} y+C_{2} z+D_{2} = 0 \\ A_{3} x+B_{3} y+C_{3} z+D_{3} = 0, & A_{4} x+B_{4} y+C_{4} z+D_{4} = 0 \end{array}

Using the ranks $r$ and $R$ of the matrices

\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \\ A_{4} & B_{4} & C_{4} \end{smallmatrix}\right] \text{ and }\left[\begin{smallmatrix} A_{1} & B_{1} & C_{1} & D_{1} \\ A_{2} & B_{2} & C_{2} & D_{2} \\ A_{3} & B_{3} & C_{3} & D_{3} \\ A_{4} & B_{4} & C_{4} & D_{4} \end{smallmatrix}\right]

respectively, express the conditions necessary and sufficient for these planes to belong to:

?

(1)

one bundle;

(2)

one proper bundle;

(3)

one improper bundle.

Problem 392

Let $\Pi_{i}(i = 1,2,3,4)$ be planes with equations $A_{i} x + B_{i} y+C_{i} z+D_{i} = 0$ , every three of which meet at one point, but which have no common point altogether. Prove that any function of the form $A x+B y+C z+D$ can be represented as a linear combination of the left-hand sides of the equations of the planes $\Pi_{i}$ , i.e., there exist constants $\lambda_{i}$ such that

A x+B y+C z+D = \sum _{i = 1}^{4} \lambda _{i}\left(A_{i} x+B_{i} y+C_{i} z+D_{i}\right)

?

Problem 393

Describe all quadruples

\Pi _{i}:\left\{ A_{i} x+B_{i} y+C_{i} z+D_{i} = 0\right\} , \quad i = 1,2,3,4

of planes in space possessing the following property: any function of the form $A x+B y+C z+D$ can be represented as a linear combination of the left-hand sides of the equations of the planes $\Pi_{i}$ , i.e., there exist constants $\lambda_{i}$ such that

A x+B y+C z+D = \sum _{i = 1}^{4} \lambda _{i}\left(A_{i} x+B_{i} y+C_{i} z+D_{i}\right)

?

Problem 394

(1)(2)

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(1)

Describe all possible families of planes whose equations $A x+B y+C z+D = 0$ can be represented as linear combinations

A x+B y+C z+D = \sum _{i = 1}^{3} \lambda _{i}\left(A_{i} x+B_{i} y+C_{i} z+D_{i}\right)

of the equations $A_{i} x+B_{i} y+C_{i} z+D_{i} = 0$ of three fixed planes $\Pi_{i}(i = 1,2,3)$ .

(2)

Do the same for four planes.

Problem 395

(1)(2)

Set up the equation of the plane containing the line of intersection of the planes $\Pi_{1}, \Pi_{2}$ and passing through the point $M$ :

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(1)

$\Pi_{1}: x+y-z-1 = 0, \quad \Pi_{2}: 2 x-3 y-3 z-2 = 0, M(1,0,-2)$ ;

(2)

$\Pi_{1}: 2 x-y+3 z-2 = 0, \Pi_{2}: x+y-2 z-1 = 0, \quad M(2,1,-3)$ .

Problem 396

Set up the equation of the plane passing through the line of intersection of the planes $x-2 y+z-1 = 0,2 x-3 y+4 z-5 = 0$ and parallel to the vector $(3,2,-2)$ .

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Problem 397

Set up the equation of the plane passing through the line of intersection of the planes $2 x-y+5 z-4 = 0, y-7 z+12 = 0$ and cutting off on the axes $O x$ and $O z$ equal nonzero segments.

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Problem 398

Show that the planes

\begin{array}{ll} \Pi _{1}: x-y+z+1 = 0, & \Pi _{2}: 2 x+3 y-z+2 = 0 \\ \Pi _{3}: x = 0, & \Pi _{4}: x+2 y-z+1 = 0 \end{array}

form a tetrahedron. Pass a plane through the line of intersection of the planes $\Pi_{1}$ and $\Pi_{2}$ parallel to the line of intersection of $\Pi_{3}$ and $\Pi_{4}$ .

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Problem 399

Write the equation of the plane passing through the point with coordinates $(2,-1,-3)$ and parallel to the lines of intersection of the planes $\Pi_{1}$ and $\Pi_{2}$ , and of the planes $\Pi_{3}$ and $\Pi_{4}$ , where the planes $\Pi_{1}, \Pi_{2}, \Pi_{3}$ and $\Pi_{4}$ are given by the equations $x+y-z+2 = 0,2 x-y+3 z-1 = 0$ , $3 x+2 y+z-4 = 0$ and $x+3 y+2 z-2 = 0$ , respectively.

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Problem 400

Write the equation of the plane parallel to the lines of intersection of the planes $\Pi_{1}, \Pi_{2}$ and $\Pi_{3}, \Pi_{4}$ and equidistant from these two lines; the planes $\Pi_{i}(i = 1,2,3,4)$ are the same as in the preceding problem.

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Problem 401

Set up the equation of the plane passing through the axis $O z$ and equidistant from the points $(2,3,-10)$ and $(1,-3,19)$ .

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Problem 402

The vertices of a tetrahedron $P(3,5,-1), Q(7,5,3), R(9,-1,5)$ , $S(5,3,-3)$ are given. Write the equations of the planes equidistant from all vertices of the tetrahedron $P Q R S$ .

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Problem 403

Set up the equation of the line that passes through the point $(1,2,-1)$ and intersects the lines

\begin{cases} x - z = 0 \\ 3 x - 2 y - 3 = 0 ; \end{cases} \quad \begin{cases} 2 x-y+1 = 0 \\ x+2 y-z+3 = 0 \end{cases}

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Problem 404

Show that the three planes $x+2 y-z-4 = 0,3 x-2 y + 3 z-6 = 0,4 y-3 z+3 = 0$ form a prism, and write the equation of the plane passing through the line of intersection of the first two faces of the prism and parallel to the third.

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Problem 405

Through the point $P(-1,2,3)$ pass a plane so that in the triangle cut off on it by the coordinate planes, the point $P$ is the point where the medians meet.

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Problem 406

Let $\Pi_{i}(i = 1,2,3)$ be planes with equations $A_{i} x+B_{i} y + C_{i} z+D_{i} = 0$ , meeting at one point, and let $P\left(x_{0}, y_{0}, z_{0}\right)$ be a point not lying in any of the planes $\Pi_{i}$ . Write the equation of the plane $\Pi$ intersecting the planes $\Pi_{i}$ and passing through the point $P$ such that the latter is the point where the medians meet of the triangle in the plane $\Pi$ , bounded by the lines of intersection of the plane $\Pi$ with the planes $\Pi_{i}$ .

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