Math Problems Vault

Problem 459

Find the planes projecting the line

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

onto the coordinate planes.

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Problem 460

Find the angles of the line

\begin{cases} 3x-y+2z = 0, \\ 6x-3y+2z = 2 \end{cases}

with the coordinate axes.

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Problem 461

Find the angle between the lines

\begin{cases} x+2y+z-1 = 0, \\ x-2y+z+1 = 0 \end{cases}

and

\begin{cases} x-y-z-1 = 0, \\ x-y+2z+1 = 0. \end{cases}

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Problem 462

Find the angle between the lines

\begin{cases} x+y = 0, \\ x-y = 2 \end{cases}

and

\begin{cases} y+z = 0, \\ y-z+2 = 0. \end{cases}

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Problem 463

The plane $x+y+z = a$ is intersected by the planes $x = y$ and $3x = y$ . Determine the angle between these lines of intersection.

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Problem 464

(a)(b)

The line

\begin{cases} x+y-z = 0, \\ y = x \tg \alpha \end{cases}

is projected:

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(a)

onto the plane $xOz$ ;

(b)

onto the plane $yOz$ .

Determine the angle between these projections.

Problem 465

Find the angle between the line

\begin{cases} x+y+3z = 0, \\ x-y-z = 0 \end{cases}

and the plane $x-y-z+1 = 0$ .

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Problem 466

Draw a line through the points $(1,2,1)$ and $(2,1,3)$ .

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Problem 467

Draw a line through the points $(1,2,1)$ and $(1,2,3)$ .

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Problem 468

Through the points $A(2,-1,3)$ and $B(0,2,-1)$ a line is drawn. Find the point of intersection of the line $AB$ with the plane $3x-y+z = 0$ .

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Problem 469

The points $A(2,1,1)$ and $B(1,2,2)$ are projected from the origin onto the plane $x+y+z = 10$ . Find the coordinates of the projections $A'$ and $B'$ and the distance $\left|A'B'\right|$ between these points.

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Problem 470

Through the point $(2,1,3)$ draw a line parallel to the plane $x+y+z = a$ and intersecting the line

\begin{cases} x = y, \\ y = 2z. \end{cases}

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Problem 471

Through the point $(1,2,1)$ draw a line perpendicular to the plane $x+2y-z = 0$ .

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Problem 472

Through the point $(2,1,1)$ draw a line parallel to the planes $2x-y+1 = 0$ , $y-1 = 0$ .

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Problem 473

Through the point $(-1,2,1)$ draw a line parallel to the line

\begin{cases} x+y-2z-1 = 0, \\ x+2y-z+1 = 0. \end{cases}

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Problem 474

Through the point $(2,1,1)$ draw a line parallel to the planes $x-y+z+2 = 0$ , $x+y+2z-1 = 0$ .

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Problem 475

Through the point $(2,2,1)$ draw a plane perpendicular to the line $x+2y-z+1 = 0$ , $2x+y-z = 0$ .

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Problem 476

Through the point $(1,1,2)$ draw a plane parallel to the lines

\dfrac {x+1}{2} = \dfrac {y-1}{1} = \dfrac {z+1}{2} \qquad \text{and} \qquad \dfrac {x+1}{1} = \dfrac {y-1}{2} = \dfrac {z+1}{1}.

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Problem 477

Through the point $(1,2,1)$ draw a plane parallel to the lines

\begin{cases} x+2y-z+1 = 0, \\ x-y+z-1 = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} 2x-y+z = 0, \\ x-y+z = 0. \end{cases}

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Problem 478

Through the line

\begin{cases} 2x-y+z-1 = 0, \\ x+y-z = 0 \end{cases}

and the point $(2,1,1)$ draw a plane.

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Problem 479

Through the line

\begin{cases} x-1 = 0, \\ x+2y-z-1 = 0 \end{cases}

draw a plane perpendicular to the plane $x+y+z = 0$ .

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Problem 480

Find the projection of the line

\begin{cases} x-y-z = 0, \\ x-5y+z = 10 \end{cases}

onto the plane $x+2y+3z = 2$ and determine the angle formed by this projection with the plane $xOy$ .

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Problem 481

Through the axis $Oz$ draw a plane making an angle of $60^{\circ }$ with the plane $2x+y-\sqrt{5}z = 7$ .

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Problem 482

Through the line $x = y = z$ draw a plane making an angle of $45^{\circ }$ with the plane $x = y$ .

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Problem 483

The line $\dfrac {x-1}{2} = \dfrac {y-2}{3} = \dfrac {z}{1}$ is projected from the point $(5,5,5)$ onto the plane $xOy$ . Find the equations of the projection.

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Problem 484

Through the point $(1,2,-1)$ draw a line intersecting both lines

\dfrac {x-1}{2} = \dfrac {y+1}{6} = \dfrac {z+3}{3} \qquad \text{and} \qquad \dfrac {x-2}{3} = \dfrac {y}{1} = \dfrac {z+3}{-1}.

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Problem 485

Through the line

\begin{cases} x+y+z = 0, \\ 2x-y+3z = 0 \end{cases}

draw a plane parallel to the line $x = 2y = 3z$ .

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Problem 486

Find the equation of the plane perpendicular to the plane $z = 0$ and passing through the perpendicular dropped from the point $(1,-1,1)$ to the line

\begin{cases} x = 0, \\ y-z+1 = 0. \end{cases}

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Problem 487

Find the equation and the length of the altitude of the triangle cut off on the plane $3x-y+4z-12 = 0$ by the coordinate planes; the altitude is dropped from the vertex lying on $Oz$ .

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Problem 488

Find the point of intersection of the line

\dfrac {x-1}{2} = \dfrac {y+1}{1} = \dfrac {z+3}{2}

with the plane $x+y-z+1 = 0$ .

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Problem 489

Find the projection of the point $(2,1,1)$ onto the plane $x+y+3z+5 = 0$ .

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Problem 490

Find the projection of the point $(2,3,1)$ onto the line

\begin{cases} x = t-7, \\ y = 2t-2, \\ z = 3t-2. \end{cases}

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Problem 491

Find the equations of the projection of the line

\begin{cases} 2x-y+z-1 = 0, \\ x+y-z+1 = 0 \end{cases}

onto the plane $x+2y-z = 0$ .

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Problem 492

Find the reflection of the point $(-1,2,0)$ in the plane $x+2y-z+1 = 0$ .

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Problem 493

Determine $\lambda$ so that the lines $\dfrac {x-1}{1} = \dfrac {y+1}{2} = \dfrac {z-1}{\lambda }$ and $\dfrac {x+1}{1} = \dfrac {y-1}{1} = \dfrac {z}{1}$ intersect.

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Problem 494

Find the lines perpendicular to the line

\begin{cases} y-z+1 = 0, \\ x+2z = 0 \end{cases}

and lying in the plane $x+y+z+1 = 0$ .

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Problem 495

Through the point of intersection of the plane $x+y+z = 2$ and the line

\begin{cases} 2y = x, \\ z = -1 \end{cases}

draw a line lying in the given plane and perpendicular to the given line.

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Problem 496

Find the equations and the length of the perpendicular dropped from the point $(0,-1,1)$ to the line

\begin{cases} y+1 = 0, \\ x+2z-7 = 0. \end{cases}

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Problem 497

The line $x = 2y = 3z$ and the point $(0,0,1)$ are given. From this point a perpendicular is dropped to the given line, and the given line itself together with the perpendicular dropped onto it are projected onto the plane $xOz$ . Find the angle between the projections.

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Problem 498

Find the point symmetric to the point $(x_{0}, y_{0}, z_{0})$ with respect to the line $\dfrac {x}{l} = \dfrac {y}{m} = \dfrac {z}{n}$ .

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Problem 499

Find the plane passing through the origin and through the perpendicular dropped from the point $(1,-1,0)$ to the line

\begin{cases} x = z+3, \\ y = -2z-3. \end{cases}

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Problem 500

Find the equations of the common perpendicular to the lines

\begin{cases} x+4z+1 = 0, \\ x-4y+9 = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} y = 0, \\ x+2z+4 = 0. \end{cases}

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Problem 501

The lines

x = y = z \qquad \text{and} \qquad \dfrac {x-1}{1} = \dfrac {y-1}{-1} = \dfrac {z}{2}

and the common perpendicular to them are projected onto the plane $xOy$ . Determine the angles between the projections.

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Problem 502

Find the plane in which the lines

\dfrac {x-1}{1} = \dfrac {y+1}{-1} = \dfrac {z-1}{2} \qquad \text{and} \qquad \dfrac {x-1}{-1} = \dfrac {y+1}{2} = \dfrac {z-1}{1}

lie.

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Problem 503

Find the plane in which the lines

\begin{cases} 2x+3y-z-1 = 0, \\ x+y-3z = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} x+5y+4z-3 = 0, \\ x+2y+2z-1 = 0. \end{cases}

lie.

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Problem 504

Find the plane in which the lines

\begin{cases} x = 2t-1, \\ y = 3t+2, \\ z = 2t-3 \end{cases} \qquad \text{and} \qquad \begin{cases} x = 2t+3, \\ y = 3t-1, \\ z = 2t+1. \end{cases}

lie.

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Problem 505

Prove that the distance from the point $M(x,y,z)$ to the line passing through the point $A(a,b,c)$ can be expressed by the formula $d = \dfrac {\left|\overrightarrow {r} \times \overrightarrow {P}\right|}{\left|\overrightarrow {P}\right|}$ , where $\overrightarrow {r} = \overrightarrow {AM}$ , and $\overrightarrow {P}$ is any vector directed along the line.

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Problem 506

Using the previous result, prove that the distance $d$ from the point $M(x_{1},y_{1},z_{1})$ to the line

\dfrac {x-a}{l} = \dfrac {y-b}{m} = \dfrac {z-c}{n}

can be expressed by the formula

d^{2} = \dfrac {1}{l^{2}+m^{2}+n^{2}}\Bigl(\bigl(m(x_{1}-a)-l(y_{1}-b)\bigr)^{2}+\bigl(n(x_{1}-a)-l(z_{1}-c)\bigr)^{2}+\bigl(n(y_{1}-b)-m(z_{1}-c)\bigr)^{2}\Bigr).

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Problem 507

Find the distance from the point $(3,-1,2)$ to the line

\begin{cases} 2x-y+z-4 = 0, \\ x+y-z+1 = 0. \end{cases}

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Problem 508

Prove that the distance between two nonparallel lines can be expressed by the formula

d = \dfrac {(\overrightarrow {r}_{1} \times \overrightarrow {r}_{2}) \cdot \overrightarrow {r}_{3}}{\left|\overrightarrow {r}_{1} \times \overrightarrow {r}_{2}\right|},

where $\overrightarrow {r}_{1}$ and $\overrightarrow {r}_{2}$ are any vectors lying on the given lines, and $\overrightarrow {r}_{3}$ is a vector going from a point on one line to a point on the other line.

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Problem 509

Prove that the length $d$ of the common perpendicular to the lines

\dfrac {x-a}{l} = \dfrac {y-b}{m} = \dfrac {z-c}{n} \qquad \text{and} \qquad \dfrac {x-a_{1}}{l_{1}} = \dfrac {y-b_{1}}{m_{1}} = \dfrac {z-c_{1}}{n_{1}}

can be represented by the formula

d = \dfrac {\begin{vmatrix} a_{1}-a & b_{1}-b & c_{1}-c \\ l & m & n \\ l_{1} & m_{1} & n_{1} \end{vmatrix}}{\sqrt{(lm_{1}-l_{1}m)^{2}+(mn_{1}-m_{1}n)^{2}+(ln_{1}-l_{1}n)^{2}}}.

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Problem 510

Find the distance between the lines

\begin{cases} x+y-z-1 = 0, \\ 2x+y-z-2 = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} x+2y-z-2 = 0, \\ x+2y+2z+4 = 0. \end{cases}

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Problem 511

Find the distance between the lines $x = y = z$ and

\begin{cases} x-1 = 0, \\ y-2 = 0. \end{cases}

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Problem 512

Prove that the distance $h$ between parallel lines can be expressed by the formula

h = \dfrac {\left|\overrightarrow {r} \times \overrightarrow {P}\right|}{\left|\overrightarrow {P}\right|},

where $\overrightarrow {r}$ is a vector going from a point on one line to a point on the other line, and $\overrightarrow {P}$ is a vector parallel to the given lines.

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Problem 513

Prove that the distance between the lines

\dfrac {x-a}{l} = \dfrac {y-b}{m} = \dfrac {z-c}{n} \qquad \text{and} \qquad \dfrac {x-a_{1}}{l} = \dfrac {y-b_{1}}{m} = \dfrac {z-c_{1}}{n}

can be expressed by the formula

h^{2} = \dfrac {1}{l^{2}+m^{2}+n^{2}}\Bigl(\bigl(m(a_{1}-a)-l(b_{1}-b)\bigr)^{2}+\bigl(n(b_{1}-b)-m(c_{1}-c)\bigr)^{2}+\bigl(l(c_{1}-c)-n(a_{1}-a)\bigr)^{2}\Bigr).

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Problem 514

Find the distance between the parallel lines

\begin{cases} x = t+1, \\ y = 2t-1, \\ z = t \end{cases} \qquad \text{and} \qquad \begin{cases} x = t+2, \\ y = 2t-1, \\ z = t+1. \end{cases}

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Problem 515

If the vectors $\overrightarrow {r}_{1}$ and $\overrightarrow {r}_{2}$ have equal length and lie on given intersecting lines, then the vectors $\overrightarrow {r}_{1} \pm \overrightarrow {r}_{2}$ are parallel to the bisectors of the angles between the lines. Starting from this remark, prove that the bisectors of the angles between the lines

\dfrac {x-a}{l} = \dfrac {y-b}{m} = \dfrac {z-c}{n} \qquad \text{and} \qquad \dfrac {x-a}{l_{1}} = \dfrac {y-b}{m_{1}} = \dfrac {z-c}{n_{1}}

can be represented by the equations

x = a+\left(\dfrac {l}{\Delta } \pm \dfrac {l_{1}}{\Delta _{1}}\right)t, \qquad y = b+\left(\dfrac {m}{\Delta } \pm \dfrac {m_{1}}{\Delta _{1}}\right)t,

z = c+\left(\dfrac {n}{\Delta } \pm \dfrac {n_{1}}{\Delta _{1}}\right)t,

where

\Delta = \sqrt{l^{2}+m^{2}+n^{2}}, \qquad \Delta _{1} = \sqrt{l_{1}^{2}+m_{1}^{2}+n_{1}^{2}}.

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Problem 516

Find the bisectors of the angles between the lines

\dfrac {x-3}{6} = \dfrac {y+2}{-3} = \dfrac {z-1}{2} \qquad \text{and} \qquad \dfrac {x-3}{14} = \dfrac {y+2}{-5} = \dfrac {z-1}{2}.

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Problem 517

Find the bisectors of the angles between the lines

\dfrac {x-1}{3} = \dfrac {y-2}{8} = \dfrac {z-3}{1} \qquad \text{and} \qquad \dfrac {x-1}{4} = \dfrac {y-2}{7} = \dfrac {z-3}{3}.

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Problem 518

Draw the bisectors of the angles between the lines

\dfrac {x}{10} = \dfrac {y}{11} = \dfrac {z}{2} \qquad \text{and} \qquad \dfrac {x}{1} = \dfrac {y}{2} = \dfrac {z}{-2}

and find the angles into which the projections of the bisectors onto the plane $xOz$ divide the projections of the angles onto the same plane.

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Problem 519

Find the lines that intersect the lines

\begin{cases} x+y-z-1 = 0, \\ x-y+z+1 = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} 2x-y+z-1 = 0, \\ x+y-z+1 = 0 \end{cases}

and are parallel to the plane $x+y+z = 0$ .

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Problem 520

Find the lines that intersect the three lines:

\begin{cases} x = 0, \\ y = 0, \end{cases} \qquad \begin{cases} x = 1, \\ z = 0, \end{cases} \qquad \begin{cases} y-1 = 0, \\ z+1 = 0. \end{cases}

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Problem 521

Prove that the planes passing through the edges of a trihedral angle and the bisectors of the opposite faces intersect along a line.

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Problem 522

Three mutually perpendicular lines are rigidly attached to a solid body. Initially they were directed along the axes $Ox$ , $Oy$ , $Oz$ , and then came to lie along the axes $Ox_{1}$ , $Oy_{1}$ , $Oz_{1}$ . The cosines of the angles between the axes are given by the table:

	$x_{1}$	$y_{1}$	$z_{1}$
$x$	$a_{1}$	$b_{1}$	$c_{1}$
$y$	$a_{2}$	$b_{2}$	$c_{2}$
$z$	$a_{3}$	$b_{3}$	$c_{3}$

Prove that such a displacement of the body could have been achieved by a rotation about a line, and find that line.

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Problem 523

The coordinates $x$ , $y$ , $z$ pass into the coordinates $x_{1}$ , $y_{1}$ , $z_{1}$ by the formulas: $9x_{1} = 4x+7y-4z$ , $9y_{1} = x+4y+8z$ , $9z_{1} = 8x-4y+z$ . Then by the same formulas the coordinates $x_{1}$ , $y_{1}$ , $z_{1}$ pass into the coordinates $x_{2}$ , $y_{2}$ , $z_{2}$ , and so on. Prove that $x_{4} = x$ , $y_{4} = y$ , $z_{4} = z$ .

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Problem 524

The coordinates $x$ , $y$ , $z$ pass into the coordinates $x_{1}$ , $y_{1}$ , $z_{1}$ by the formulas:

9x_{1} = (5\cos \varphi +4)x+(-4\cos \varphi +3\sin \varphi +4)y+(-2\cos \varphi +6\sin \varphi +2)z,

9y_{1} = (-4\cos \varphi -3\sin \varphi +4)x+(5\cos \varphi +4)y+(-2\cos \varphi +6\sin \varphi +2)z,

9z_{1} = (-2\cos \varphi +6\sin \varphi +2)x+(-2\cos \varphi -6\sin \varphi +2)y+(8\cos \varphi +1)z.

Then by the same formulas the coordinates $x_{1}$ , $y_{1}$ , $z_{1}$ pass into the coordinates $x_{2}$ , $y_{2}$ , $z_{2}$ , and so on. Prove that the coordinates $x_{n}$ , $y_{n}$ , $z_{n}$ are expressed in terms of $x$ , $y$ , $z$ by the same formulas as $x_{1}$ , $y_{1}$ , $z_{1}$ , except that the angle $\varphi$ is replaced by $n\varphi$ .

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