Math Problems Vault

Problem 6.45

(1)(2)

Find a normal vector of the plane:

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(1)

$Ax + By + Cz + D = 0$

(2)

$x = x_{0} + a_{1} u + a_{2} v, y = y_{0} + b_{1} u + b_{2} v, z = z_{0} + c_{1} u + c_{2} v$ .

Problem 6.46

Find the equations of the line passing through the point $A(1, - 1, 2)$ and perpendicular to the plane:

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(1)

$x - 3y + 2z + 1 = 0$

(2)

$x = 5$

(3)

$y = 4$

(4)

$z = 3$

(5)

$x = 4 - u + v, \quad y = 2 + u + 2v, \quad z = - 1 + 7u + 3v$ .

Problem 6.47

Find the equation of the plane passing through the point $A(1, 3, 1)$ and perpendicular to the line:

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(1)

$x + y - z + 2 = 0, \quad 2x + 3y + z - 1 = 0$

(2)

$\frac{x + 1}{3} = \frac{y - 2}{4} = \frac{z + 2}{21}$

(3)

$x = 2, \quad y = 3$

(4)

$x = 0, z = 0$

(5)

$y = - 1, \quad z = 2$ .

Problem 6.48

Find the equation of the plane passing through the point $A(2, 1, - 1)$ and perpendicular to the two planes: $x - y + 5z + 1 = 0$ and $2x + y = 3$ .

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Problem 6.49

(1)(2)

Find the equation of the plane perpendicular to the plane $x + 3y - z + 2 = 0$ and passing through the line:

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(1)

$\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{4}$

(2)

$2x - y + z = 0, x + 2y + z - 3 = 0$ .

Problem 6.50

In the pencil determined by the planes $x + 2y - 3z + 5 = 0$ and $4x - y + 3z + 5 = 0$ , find two mutually perpendicular planes, one of which passes through the point $M(1, 3, 1)$ .

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Problem 6.51

Point $A$ lies on the line $\frac{x - 3}{2} = \frac{y}{3} = \frac{z - 1}{ - 1}$ , and $A$ is equidistant from the points $B(3, 0, - 2)$ and $C(- 1, 1, 5)$ . Find the coordinates of point $A$ .

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Problem 6.52

Find the distance from the point $A(3, 1, - 1)$ to the plane:

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(1)

$x - y - 5z + 2 = 0$

(2)

$x - 2y + 2z - 2 = 0$

(3)

$x - 2y + 2z + 7 = 0$

(4)

$x - 2y + 2z = 0$

(5)

$x - 2y + 2z + 1 = 0$

(6)

$x = 1$

(7)

$y = 5$

(8)

$z = 0$

Problem 6.53

(1)(2)(3)

Find the distance between the parallel planes:

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(1)

$6x - 3y + 2z + 5 = 0$ and $6x - 3y + 2z - 9 = 0$

(2)

$2x + 2y - z + 3 = 0$ and $2x + 2y - z + 18 = 0$

(3)

$3x + 4z + 1 = 0$ and $6x + 8z - 1 = 0$ .

Problem 6.54

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

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(1)

Find the equations of the planes parallel to the plane $6x - 3y + 2z + 5 = 0$ and at distance 3 from it.

(2)

Find the equations of the planes parallel to the plane $x + 3y - z + \sqrt{11} = 0$ and at distance 3 from it.

(3)

Find the equations of the planes parallel to the plane $2x + 2y - z + 3 = 0$ and at distance 3 from the point $A(1, 2, - 1)$ .

(4)

Find the equations of the planes parallel to the plane $3x + 4z + 1 = 0$ and at distance 3 from the origin.

Problem 6.55

Point $A$ lies on the line $\frac{x - 1}{2} = \frac{y}{3} = \frac{z + 1}{1}$ . The distance from point $A$ to the plane $x + y + z + 3 = 0$ equals $\sqrt{3}$ . Find the coordinates of point $A$ .

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Problem 6.56

Point $A$ lies on the line $\frac{x - 1}{1} = \frac{y - 3}{3} = \frac{z + 4}{ - 5}$ , and $A$ is equidistant from the point $B(0, 1, 1)$ and from the plane $2x - y + 2z + 1 = 0$ . Find the coordinates of point $A$ .

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Problem 6.57

Points $A(1, - 1, 2)$ and $B(3, 0, 4)$ are vertices of a cube $ABCDA_{1} B_{1} C_{1} D_{1}$ . The vector $\overrightarrow {AD}$ is perpendicular to the line $x = 0, y - z = 0$ , and the orientation of the triple of vectors $\overrightarrow {AB}, \overrightarrow {AD}$ . $\overrightarrow {AA_{1}}$ coincides with the orientation of the triple of basis vectors of the coordinate system, and the sum of the coordinates of the vector $\overrightarrow {AA_{1}}$ is negative. Find the equations of the faces of the cube.

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Problem 6.58

Points $A(- 3, 0, 0)$ and $B(3, 0, 0)$ are vertices of a regular tetrahedron $ABCD$ , the vertex $C$ is at distance $3\sqrt{2}$ from the coordinate plane $Oxy$ , and all coordinates of the point $C$ are nonnegative. The orientation of the triple of vectors $\overrightarrow {AB}, \overrightarrow {AC}, \overrightarrow {AD}$ coincides with the orientation of the triple of basis vectors of the coordinate system. Find the equations of the faces of the tetrahedron.

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Problem 6.59

(1)(2)(3)

Given the point $A(3, - 1, 1)$ . Find:

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(1)

the coordinates of the projections of the point $A$ onto the coordinate planes and the coordinates of the points symmetric to $A$ with respect to the coordinate planes

(2)

the coordinates of the projection of the point $A$ onto the plane $x + 2y + 2z + 6 = 0$ and the coordinates of the point symmetric to $A$ with respect to this plane

(3)

the coordinates of the projection of the point $A$ onto the plane $2x + 3y + 6z + 40 = 0$ and the coordinates of the point symmetric to $A$ with respect to this plane.

Problem 6.60

Find the equations of the line symmetric to the line $\frac{x - 2}{3} = \frac{y + 1}{1} = \frac{z - 2}{4}$ with respect to the plane $5x - y + z - 4 = 0$ .

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Problem 6.61

(1)(2)(3)

Find the equations of the projections onto the plane $x + 5y - z - 25 = 0$ of the following lines:

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(1)

$\frac{x + 1}{4} = \frac{y}{2} = \frac{z - 1}{3}$

(2)

$x - y + 2z - 1 = 0, \quad 3x - y + 2z + 2 = 0$

(3)

$\frac{x + 1}{1} = \frac{y}{5} = \frac{z - 1}{ - 1}$ .

Problem 6.62

Find the angle between the planes:

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(1)

$x + 4y - z + 1 = 0$ and $x + y - z - 3 = 0$

(2)

$x + 2y - z = 1$ and $x - y = 3$

(3)

$x + 2y - 2z = 0$ and $z = 5$

(4)

$x + 2y - z - 1 = 0$ and $3x - 5y - 7z = 0$

(5)

$x + 3y - z + 1 = 0$ and $x = 1 - u, \quad y = 2 - 3u - v, \quad z = 7 + u + v$

(6)

$x - 3y + 2z + 1 = 0$ and $6z - 9y + 3x + 5 = 0$ .

Problem 6.63

(1)(2)(3)

Find the angle between the lines:

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(1)

$2x + y - z + 1 = 0, x + 3y + z + 2 = 0$ and $x + 3y - z + 2 = 0$ , $x + y + z - 1 = 0$

(2)

$\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z + 3}{ - 1}$ and $\frac{x + 1}{ - 3} = \frac{y}{4} = \frac{z - 10}{6}$

(3)

$x = 5 - 2t, \quad y = 6 + 4t, \quad z = 8t$ and $x = 1 + t, \quad y = - 2t, \quad z = 3 - 4t$ .

Problem 6.64

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

Find the angle between the plane $4x + 4y - 7z + 1 = 0$ and the line:

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(1)

$x + y + z + 1 = 0, 2x + y + 3z + 2 = 0$

(2)

$\frac{x - 1}{3} = \frac{y + 2}{2} = \frac{z}{ - 6}$

(3)

$\frac{x - 2}{4} = \frac{y - 1}{4} = \frac{z + 3}{ - 7}$

(4)

$\frac{x - 1}{11} = \frac{y + 1}{ - 4} = \frac{z + 3}{4}$ .

Problem 6.65

(1)(2)

Find the equations of the line passing through the point $A(1, 3, 2)$ parallel to the plane $Oxy$ and making:

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(1)

an angle of $45^{\circ }$ with the line $x = y, z = 0$

(2)

an angle of $\arcsin (1 / \sqrt{10})$ with the plane $x - y = 1$ .

Problem 6.66

Find the equation of the plane passing through the point $A(- 1, 2, 1)$ parallel to the line $\frac{x}{2} = - \frac{y}{3} = - z$ and making an angle of $60^{\circ }$ with the line $x = y, z = 0$ .

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Problem 6.67

Find the equation of the plane passing through the line $x + 5y + z = 0, \quad x - z + 4 = 0$ and making an angle $45^{\circ } \mathrm{c}$ the plane $x - 4y - 8z + 1 = 0$ .

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Problem 6.68

(1)(2)(3)

Prove that the two given lines intersect, and find the equations of the bisectors of the acute and obtuse angles between them:

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(1)

$x = 4 - 4t, y = 1 + 4t, z = - 5 + 7t$ and $x = - 3 + t, y = - 1 + 2t, \quad z = - 4 + 2t$

(2)

$x = 4 + t, y = 1 - t, \quad z = 5 + 4t$ and $x = - 3 - 3t, \quad y = 8 + 3t$ , $z = 1$

(3)

$x = 1 + 2t, \quad y = 2 + 3t, \quad z = 11 - 6t$ and $x = 1 + t, \quad y = 3 + t$ , $z = 7 - t$ .

Problem 6.69

The legs of an isosceles triangle have a common vertex $A(3, 4, 5)$ , the other two vertices lie on the axes $Ox$ and $Oy$ , and the plane of the triangle is parallel to the axis $Oz$ . Find the angles of the triangle and find the equation of its plane.

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Problem 6.70

(1)(2)(3)

Given a point $A(2, - 1, 0)$ and a line $l$ . Compute the distance from the point $A$ to the line $l$ ; find the coordinates of the projection of the point $A$ onto $l$ and the coordinates of the point $B$ symmetric to $A$ with respect to $l$ ; find the equations of the line passing through the point $A$ and meeting the given line at a right angle ("drop a perpendicular" from the point $A$ to $l$ ). The line $l$ is given by the equations:

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(1)

$\frac{x - 7}{3} = \frac{y - 1}{4} = \frac{z - 3}{2}$

(2)

$x = 1 + 2t, \quad y = 2 - 2t, \quad z = - 3 + t$

(3)

$2x + y - z + 1 = 0, x + y + z + 2 = 0$ .

Problem 6.71

The point $A$ lies on the line $x - y - 3 = 0, 2y + z = 0$ . The distance from the point $A$ to the line $x = y = z$ equals $\sqrt{6}$ . Find the coordinates of the point $A$ .

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Problem 6.72

(1)(2)(3)

Find the distance between the lines:

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(1)

$\frac{x - 4}{3} = \frac{y + 1}{6} = \frac{z - 1}{ - 2}$ and $\frac{x - 5}{ - 6} = \frac{y}{ - 12} = \frac{z}{4}$

(2)

$x = 3 + 2t, y = 10 - 3t, z = 3 + 4t$ and $x = 1 + 3t, y = 1 - 2t$ , $z = 1 + 3t$

(3)

$x + y + z - 1 = 0, x + 3y - z + 2 = 0$ and $x + 3y + z + 2 = 0$ , $x + 2y - z + 1 = 0$ .

Problem 6.73

(1)(2)(3)

Given lines $l_{1}$ and $l_{2}$ . Find the equations of their common perpendicular (i.e., the line meeting $l_{1}$ and $l_{2}$ at a right angle); find the points of intersection of the common perpendicular with the given lines; compute the distance between $l_{1}$ and $l_{2}$ . The lines are given by the equations:

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(1)

$x = 5 + t, \quad y = 3 - t, \quad z = 13 + t$ and $x = 6 + t, \quad y = 1 + 2t, z = 10 - t$ ;

(2)

$2x + 7y - 13 = 0, \quad 3y - 2z - 1 = 0$ and $x + y - 8 = 0, 2x + y - z = 0$ ;

(3)

$\frac{x - 6}{1} = \frac{y - 1}{2} = \frac{z - 10}{ - 1}$ and $\frac{x + 4}{ - 7} = \frac{y - 3}{2} = \frac{z - 4}{3}$ .

Problem 6.74

Points $A(- 1, - 3, 1), B(5, 3, 8), C(- 1, - 3, 5), D(2$ , $1, - 4)$ are vertices of a tetrahedron. Find:

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(1)

the length of the altitude of the tetrahedron dropped from the vertex $D$ to the face $ABC$

(2)

the length of the altitude of the base $ABC$ dropped from the vertex $C$ to the side $AB$

(3)

the distance between the skew edges $AD$ and $BC$

(4)

the angle between the skew edges $AD$ and $BC$

(5)

the angle between the edge $AD$ and the face $ABC$ .

Problem 6.75

(1)(2)(3)

The length of the edge of the cube $ABCDA_{1} B_{1} C_{1} D_{1}$ equals 1. Find:

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(1)

the distance from the vertex $A$ to the plane $B_{1} CD_{1}$

(2)

the distance between the diagonal of the cube $AC_{1}$ and the diagonal of the lateral face $CD_{1}$ skew to it

(3)

the ratios in which the points of intersection of the common perpendicular to the lines $AC_{1}$ and $CD_{1}$ with these lines divide the segments $AC_{1}$ and $CD_{1}$ .

Problem 6.76

Three faces $ABCD, ABB_{1} A_{1}$ and $ADD_{1} A_{1}$ of a parallelepiped $ABCDA_{1} B_{1} C_{1} D_{1}$ lie respectively in the planes $2x + 3y + 4z + 8 = 0, x + 3y - 6 = 0, z + 5 = 0$ ; the vertex $C_{1}$ has coordinates $6, - 5, 1$ . Find:

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(1)

the distance from the vertex $A_{1}$ to the plane $B_{1} BD$

(2)

the distance from the vertex $D$ to the line $AB$

(3)

the distance between the lines $AC$ and $A_{1} C_{1}$

(4)

the distance between the lines $AA_{1}$ and $BC$

(5)

the angle between the lines $AC$ and $C_{1} D_{1}$

(6)

the angle between the planes $BDD_{1}$ and $ACC_{1}$

(7)

the angle between the line $CA_{1}$ and the plane $DCC_{1}$ .

Problem 6.77

(1)(2)

Find the necessary and sufficient conditions for the one of the four dihedral angles, formed by two intersecting non-perpendicular planes $A_{1} x + B_{1} y + C_{1} z + D_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} z + D_{2} = 0$ , that contains the point $M_{0}\left(x_{0}, y_{0}, z_{0}\right)$ , to be:

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(1)

acute

(2)

obtuse.

Problem 6.78

Given two planes $x + 2y + 2z = 0$ and $7x + 4y + 4z = 0$ . A third plane $m$ passes through the origin $O$ so that the endpoint of its normal vector, laid off from the point $O$ , lies in the obtuse dihedral angle formed by the given planes. The cosines of the acute dihedral angles between $m$ and the given planes equal $2 / 15$ and 4 / 45 respectively. Find the equation of the plane $m$ .

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Problem 6.79

Consider the dihedral angle between the planes $x - 2y + z + 3 = 0$ and $x + y + 2z = 1$ , inside which the point $A(- 1, 0, 0)$ lies. Prove that the set of points lying inside this angle and at distances $\sqrt{6}$ and $2\sqrt{6}$ from the given planes respectively is a line. Find the equations of this line.

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Problem 6.80

Find the equation of the bisector plane of the dihedral angle between the planes $x - z - 5 = 0$ and $3x + 5y + 4z = 0$ inside which the point $A(1, 1, 1)$ lies.

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Problem 6.81

Find the equation of the bisector plane of the acute dihedral angle between the planes $x - z - 5 = 0$ and $3x + 5y + 4z = 0$ .

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Problem 6.82

(1)(2)

The faces of a tetrahedron are given by the equations $x + 2y - 2z + 3 = 0, 4x - 4y + 7z - 9 = 0, 8x + 4y + z - 3 = 0, y - z = 0$ . Find the equations of:

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(1)

the bisector plane of the interior dihedral angle between the first two faces

(2)

the line lying in the interior trihedral angle between the first three faces, all points of which are equidistant from these three faces.

Problem 6.83

The vertices of a tetrahedron are the points $A(1, 2, 3)$ , $B(- 2, 8, 9), C(5, 0, 7), D(3, 4, 2)$ . Find the radii and the coordinates of the centers of the inscribed and circumscribed spheres.

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Problem 6.84

Find the radius and the coordinates of the center of the sphere passing through the point $A(0, 1, 0)$ and tangent to the planes $x + y = 0$ , $x - y = 0, \quad x + y + 4z = 0$ .

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Problem 6.85

Find the coordinates of the center $O$ and the radius $r$ of the sphere tangent to the planes $5x - y + z - 17 = 0$ and $x + y - z + 11 = 0$ and passing through the points $A(- 7, - 1, - 1)$ and $B(1, 1, 1)$ .

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Problem 6.86

Find the coordinates of the center $O$ and the radius $r$ of the sphere tangent to the plane $x + 5y + z - 33 = 0$ and passing through the points $A(2, 3, - 2), B(- 2, 3, 4)$ and $C(0, - 1, 2)$ .

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Problem 6.87

The vertices of a triangle are the points $A(1, 2, 3)$ , $B(1, 5, - 1), C(5, 3, - 5)$ . Find the radii and the coordinates of the centers of the inscribed and circumscribed circles.

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Problem 6.88

(1)(2)

Prove that the three planes $x - 2y + 2z + 3 = 0$ , $2x + 2y + z - 6 = 0, 5x + 14y - 2z - 21 = 0$ have no common points, but the three lines formed by the intersection of each pair of these planes are parallel, i.e., the planes form a prism. Find the radius and the equations of the axis of the right circular cylindrical surface:

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(1)

inscribed in this prism;

(2)

circumscribed about it.

Problem 6.89

In a regular quadrilateral pyramid $SMNPQ (S$ is the apex), points $H$ and $F$ are the midpoints of the edges $MN$ and $NP$ respectively. The point $E$ lies on the segment $SH$ , with $\left|SH\right| = 3$ , $\left|SE\right| = 9 / 4$ . The distance from the point $S$ to the line $EF$ equals $\sqrt{5}$ . Find the volume of the pyramid.

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Problem 6.90

The base of a right prism $ABCDA_{1} B_{1} C_{1} D_{1}$ is a rhombus $ABCD$ with angle $\angle A = 60^{\circ }$ . The length of the side of the base of the prism equals $a$ , the length of the lateral edge equals $\sqrt{3} a$ . The point $E$ is the orthogonal projection of the vertex $C_{1}$ onto the plane $AB_{1} D_{1}$ , and the point $F$ is the orthogonal projection of the point $E$ onto the plane $AA_{1} D_{1} D$ . Find the volume of the pyramid $ADEF$ .

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Problem 6.91

In a regular prism $ABCA_{1} B_{1} C_{1}$ the length of the lateral edge equals 3. The point $M$ is the midpoint of the edge $AC$ , the point $N$ lies on the edge $B_{1} C_{1}$ , and the point $P$ belongs to the face $AA_{1} B_{1} B$ and is at distance 1 from the plane $ABC$ . It is known that each of the lines $PM$ and $PN$ makes an angle of $30^{\circ }$ with the plane $AA_{1} B_{1} B$ , and the line $PN$ makes an angle of $30^{\circ }$ with the plane $BB_{1} C_{1} C$ . Find the volume of the prism.

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Problem 6.92

In a regular prism $ABCA_{1} B_{1} C_{1}$ the length of the side of the base equals $2a$ , the length of the lateral edge equals $a$ . Through the vertex $A$ a plane is drawn perpendicular to the line $AB_{1}$ , through the vertex $B$ a plane perpendicular to the line $BC_{1}$ , and through the vertex $C$ a plane perpendicular to the line $CA_{1}$ . Find the volume of the polyhedron bounded by these three planes and the plane $A_{1} B_{1} C_{1}$ .

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