Lines and Planes in Space

Write the general equation of a plane from its parametric equations: $x = u-6 v, y = 1+2 u, z = 2+3 v$.

Write the canonical equation of the line of intersection of the planes $x+2 y-z+2 = 0$ and $2 x+y+z+1 = 0$.

A point $M(1,2,3)$ is given. Write the equations of:

Write the equation of the plane passing through the point $(1,2,-3)$ and parallel to the vectors $(1,-1,2)$ and $(2,3,-1)$.

Write the equation of the plane passing through the points $(4,3,2),(5,2,4)$ and $(1,-1,1)$.

Find the parametric equations of the plane passing through the points $(-2,1,1),(1,2,5)$ and $(-1,2,3)$, and write its general equation.

Write the equation of the plane passing through three noncollinear points $M_{i}\left(x_{i}, y_{i}, z_{i}\right), i = 1,2,3$.

Write the equation of the plane passing through the point $(3,5,1)$ and cutting off on the coordinate axes segments equal in length. The coordinate system is rectangular.

Find the equation of the plane passing through the points $(1,2,1)$ and $(3,1,1)$ and parallel to the vector $(-1,1,2)$.

Write the equation of the plane passing through the points $M_{1}$ and $M_{2}, M_{i}\left(x_{i}, y_{i}, z_{i}\right)$, and parallel to the vector $\mathbf{v} = (\alpha , \beta , \gamma )$, not collinear with the vector $\overrightarrow {M_{1} M_{2}}$.

Set up the equations of the line that passes through the point $A(1,-5,3)$ and makes angles with the coordinate axes equal to $\pi / 3, \pi / 4,2 \pi / 3$, respectively. (The coordinate system is rectangular.)

Find a direction vector of the line of intersection of the planes $x-y+2 z-1 = 0$ and $2 x+3 y-z+5 = 0$.

Find the parametric equations of the line of intersection of the plane $2 x-y+3 z-4 = 0$ with the coordinate plane $O x y$.

Set up the parametric equations of the line passing through the point $(2,3,1)$ parallel to the plane $x-y-1 = 0$ and intersecting the axis $O y$.

Set up the equation of the plane passing through the point $M$ and containing the line $\ell$:

Find the equation of the plane containing the line $\ell$ and parallel to the vector $\mathbf{a}$, if:

Find the equation of the plane parallel to the line $\ell$ and passing through the points $A$ and $B$, if:

Set up the equations of the line lying in the plane $x+z = 0$ and intersecting the line

but having no common points with the line $x = 1+t, y = 3+4 t, z = -1-t$.

Pass a plane through the parallel lines

In the plane passing through the three points $M(1,1,-1)$, $N(-1,-1,-1), P(0,1,0)$, an affine coordinate system is chosen with origin at the point $M$ and basis vectors $\overrightarrow {M N}, \overrightarrow {M P}$. Find:

In the plane $2 x+3 y-4 z+12 = 0$ an affine coordinate system $u, v$ is chosen, whose origin is at the point $S$ of intersection of this plane with the axis $O z$, and whose basis vector endpoints are at the points $P, Q$ of intersection of the plane with the axes $O x$ and $O y$, respectively.

Three lines are given

Write the equations of the line intersecting the first two of the given lines and parallel to the third.

Write the equations of the line passing through the point $(1,2,3)$ and intersecting the lines $\frac{x}{2} = \frac{y+1}{-2} = \frac{z-2}{1}, \frac{x}{4} = \frac{y+2}{0} = \frac{z}{3}$.

Set up the equations of the projection of the line $\frac{x-2}{3} = \frac{y-1}{-2} = \frac{z}{1}$ from the point $(1,2,1)$ onto the plane $y-2 z+4 = 0$.

Prove that the six planes, each of which passes through an edge of a tetrahedron and the midpoint of the opposite edge, meet at one point.

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:

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:

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):

Two planes are given

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

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

Two planes are given by their parametric equations

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

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

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

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

Two lines are given

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

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

Two lines are given

Find the necessary and sufficient conditions for them to:

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

Three planes are given:

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

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:

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

Under what necessary and sufficient condition do the four planes

form a tetrahedron?

Four planes are given

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

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

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

Describe all quadruples

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

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

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 $(3,2,-2)$.

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.

Show that the planes

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 $\Pi_{4}$.

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.

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.

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

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 $P Q R S$.

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

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.

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.

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 $\Pi_{i}$.

Which two of the three points $A(0,2,-1), B(0,-3,2), C(-3,2,1)$ lie on the same side of the plane $3 x+2 y+z+1 = 0$?

Which of the two planes $x+2 y+4 z-11 = 0$ and $x-y+3 z-9 = 0$ separates the points $A(1,3,-1)$ and $B(5,-3,1)$?

Two points $A(1,3,-1)$ and $B(5,3,1)$ and a plane $x-4 y + 2 z-7 = 0$ are given. Determine whether the given plane intersects the segment $A B$, its extension beyond the point $A$, or beyond the point $B$.

Two points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ and a plane $A x + B y+C z+D = 0$ are given. Without finding its point of intersection with the line $P Q$, find the ratio in which this point divides the segment $P Q$.

Two points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ are given. Find in what ratio the point of intersection of the line $P Q$ with the plane

divides the segment $P Q$.

Prove that the vector with coordinates $(A, B, C)$ is always directed toward the "positive" half-space relative to the plane $A x+B y+C z+D = 0$.

A point $\left(x_{0}, y_{0}, z_{0}\right)$ is given. Under what necessary and sufficient condition does this point lie between the two parallel planes $A x+B y+C z+D = 0$ and $A x+B y+C z+E = 0$?

Find the necessary and sufficient condition for a point to lie inside the tetrahedron formed by the planes

Four planes are given: $x+y = 0, x-z = 0, x+y+2 z = 0$, $y = 0$. In each of the following cases, find planes passing through the point $P$ such that the quadrilaterals cut out on them by these planes are parallelograms, and determine whether the point $P$ lies inside or outside these parallelograms:

Find the equation of the plane that passes through the point $(2,3,-1)$ and is perpendicular to the vector $(1,-3,-1)$.

Find the equation of the plane that passes through the point $(1,1,1)$ and is perpendicular to the line

Set up the equation of the plane passing through the point $\left(x_{1}, y_{1}, z_{1}\right)$ and perpendicular to the line

Find the equation of the plane perpendicular to the line $x + 2 y-z+5 = 0,3 x+y+z-7 = 0$ and passing through the point $(2,-3,1)$.

Find the equation of the plane that is perpendicular to the planes $2 x+y-z+7 = 0$ and $x-3 y+z-1 = 0$ and passes through the point $(-1,1,0)$.