Equation of a Plane in Space
A tripod never wobbles, but a four-legged table can. Why are three points always enough to fix a plane? A line needs a point and one direction vector. A plane, which is a surface, needs a point and two directions, or else a point and one vector perpendicular to it (the normal vector). From this come several ways of writing its equation, all equivalent.
Vector and Parametric Equations
It is the same idea as for a line: you move from $A$ some amount in the direction of $\vec u$ and some amount in the direction of $\vec v$, and with that you reach the whole plane.
General Equation: Point and Normal Vector
Why? A point $X$ is in the plane if and only if $\vec{PX}$ is perpendicular to $\vec n$, that is, if the dot product $\vec n\cdot\vec{PX}=0$. Writing it in coordinates gives the equation.
Common cases:
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If a line has direction vector $\vec d$, the plane perpendicular to it has $\vec d$ as its normal vector.
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Two parallel planes share (or have proportional) normal vectors; only $D$ changes.
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If you are given parametric equations, the normal is the cross product $\vec n=\vec u\times\vec v$.
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A point $Q$ belongs to the plane if the equation holds when you substitute its coordinates.
Given the equation of the plane representing one of the walls of a room, $\pi \equiv 2x - 3y + z - 6 = 0$, determine, giving reasons, whether the following points belong…
Given the equation of the plane representing one of the walls of a room, $\pi \equiv 2x - 3y + z - 6 = 0$, determine, giving reasons, whether the following points belong to this wall:
- a) $Q(3, 0, 0)$
- b) $R(1, 1, 1)$
An engineer is designing the base of a component using CAD software. The base must be a plane passing through point $P(2, -1, 3)$ with a normal vector…
An engineer is designing the base of a component using CAD software. The base must be a plane passing through point $P(2, -1, 3)$ with a normal vector $\vec{n} = (3, 2, -4)$. Calculate the general equation of the plane that defines this base.
The surface of a solar panel on a satellite can be represented by the following parametric equations:…
The surface of a solar panel on a satellite can be represented by the following parametric equations:
$\begin{cases} x = 1 + \lambda + 2\mu \\ y = 2 - \lambda \\ z = 3\mu \end{cases}$
Obtain the general or implicit equation of this plane.
The Plane Through Three Points
Coplanar points. Four points are coplanar if the fourth satisfies the equation of the plane through the other three. If the equation contains a parameter $k$, substituting the fourth point gives an equation for $k$.
In a virtual reality scenario, three sensors are located at points $A(1, 0, 0)$, $B(0, 2, 0)$ and $C(0, 0, 3)$. Determine the general equation of the plane containing…
In a virtual reality scenario, three sensors are located at points $A(1, 0, 0)$, $B(0, 2, 0)$ and $C(0, 0, 3)$. Determine the general equation of the plane containing these three sensors.
An architect is designing a flat roof for a small pavilion. The roof must rest on three pillars whose upper ends are located at the points $A(2, 0, 3)$, $B(0, 4, 3)$ and…
An architect is designing a flat roof for a small pavilion. The roof must rest on three pillars whose upper ends are located at the points $A(2, 0, 3)$, $B(0, 4, 3)$ and $C(1, 1, 5)$, where the units are in meters.
- a) Obtain the general or implicit equation of the plane containing the roof.
- b) Calculate the points of intersection of this plane with the coordinate axes to determine where the anchors would be located if the roof extended to touch the main walls (axes).
Find the value of the parameter $k \in \mathbb{R}$ so that the points $A(1, 0, 1)$, $B(0, 2, 1)$, $C(2, 1, 0)$ and $D(k, k, 2)$ are coplanar. Once $k$ is found, write…
Find the value of the parameter $k \in \mathbb{R}$ so that the points $A(1, 0, 1)$, $B(0, 2, 1)$, $C(2, 1, 0)$ and $D(k, k, 2)$ are coplanar. Once $k$ is found, write the equation of the plane containing them in parametric form.
Plane Through a Line and a Point, or Through Two Lines
Most often you are asked for a plane built from lines. The strategy is always the same: get a point and two direction vectors (or a normal).
| Given | Point | Vectors in the plane |
|---|---|---|
| Line $r$ and external point $P$ | $P$ (or one on $r$) | $\vec d_r$ and $\vec{AP}$, with $A\in r$ |
| Two intersecting lines | the intersection point | $\vec d_r$ and $\vec d_s$ |
| Line $r$ and parallel to another line $s$ | a point of $r$ | $\vec d_r$ and $\vec d_s$ |
For the plane to be determined, the two vectors must not be proportional (that is why, in the last case, $\vec d_s$ cannot be proportional to $\vec d_r$). For two intersecting lines, first verify that they really do meet: solve the system that equates their coordinates.
Determine the equation of the plane containing the line $r \equiv \begin{cases} x = 2 - t \\ y = t \\ z = 1 + 2t \end{cases}$ and also passing through the external point…
Determine the equation of the plane containing the line $r \equiv \begin{cases} x = 2 - t \\ y = t \\ z = 1 + 2t \end{cases}$ and also passing through the external point $P(1, 1, 1)$.
Determine the equation of the plane $\pi$ containing the line $r$ given by: $$r \equiv \begin{cases} x = 1 + 2\lambda \\ y = -1 + \lambda \\ z = 3\lambda \end{cases}$$…
Determine the equation of the plane $\pi$ containing the line $r$ given by:
$$r \equiv \begin{cases} x = 1 + 2\lambda \\ y = -1 + \lambda \\ z = 3\lambda \end{cases}$$and parallel to line $s$, which passes through points $P(0, 2, 1)$ and $Q(1, 2, 4)$. Justify why the direction vector of $s$ cannot be proportional to that of $r$ in order for the plane to be uniquely determined.
Find the equation of the plane containing the following lines, first verifying that they intersect at a point: $$r \equiv \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}…
Find the equation of the plane containing the following lines, first verifying that they intersect at a point:
$$r \equiv \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \quad , \quad s \equiv \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-1}$$If a third laser beam follows the direction of the normal vector to this plane and originates from the point of intersection of $r$ and $s$, what would its vector equation be?
Intersection of a Line With a Plane
A laser beam leaving a point and hitting a flat mirror, a drone passing through a screen: in all these cases the question is where the line meets the plane.
If the line is given in symmetric form, set the fractions equal to $t$ to get the parametric equations. If it is given as the intersection of two planes, the intersection point with a third plane is the solution of the system of three equations.
In a rescue mission, a drone moves along a straight trajectory given by the line $r \equiv \frac{x-1}{2} = \frac{y}{3} = \frac{z+2}{-1}$. The drone must emit a radio…
In a rescue mission, a drone moves along a straight trajectory given by the line $r \equiv \frac{x-1}{2} = \frac{y}{3} = \frac{z+2}{-1}$. The drone must emit a radio frequency signal that is only effective if it hits a flat safety screen perpendicularly.
- a) Determine the equation of the plane representing this screen if it must pass through point $P(2, 2, 2)$ and be perpendicular to the drone's trajectory.
- b) Calculate the exact point where the drone's trajectory would hit the screen.
A laser beam starts from the point $A(1, 2, -2)$ and travels in the direction of the vector $\vec{u} = (1, -1, 1)$ until it strikes a plane mirror containing the line…
A laser beam starts from the point $A(1, 2, -2)$ and travels in the direction of the vector $\vec{u} = (1, -1, 1)$ until it strikes a plane mirror containing the line $s: \begin{cases} x - y + z = 2 \\ 2x + y - z = 1 \end{cases}$ and the point $B(0, 0, 4)$.
- a) Calculate the equation of the plane $\pi$ where the mirror is located.
- b) Find the coordinates of the point of impact $P$ of the laser on the mirror.
Intercept Equation
Volume of the tetrahedron. The plane and the three coordinate planes enclose a tetrahedron with vertices $O$, $(a,0,0)$, $(0,b,0)$, $(0,0,c)$. Its volume is
$$V=\frac{abc}{6}$$(one sixth of the volume of the rectangular prism with edges $a, b, c$).
An architect uses the "intercept equation" to define a sloping roof that intersects the coordinate axes at points $(4, 0, 0)$, $(0, 3, 0)$ and $(0, 0, 2)$. Write its…
An architect uses the "intercept equation" to define a sloping roof that intersects the coordinate axes at points $(4, 0, 0)$, $(0, 3, 0)$ and $(0, 0, 2)$. Write its equation directly and convert it to the general form $Ax + By + Cz + D = 0$.
A plane $\pi$ intersects the coordinate axes at points $X(a, 0, 0)$, $Y(0, b, 0)$ and $Z(0, 0, c)$. It is known that the plane passes through the point $P(1, 2, 2)$ and…
A plane $\pi$ intersects the coordinate axes at points $X(a, 0, 0)$, $Y(0, b, 0)$ and $Z(0, 0, c)$. It is known that the plane passes through the point $P(1, 2, 2)$ and that the distances from the origin to the intersection points are equal ($a = b = c$).
- a) Write the general equation of the plane.
- b) Calculate the volume of the tetrahedron formed by the origin and the three intersection points with the axes.
The Pencil of Planes
Why does it work? If a point of $r$ makes both $\pi_1$ and $\pi_2$ equal to zero, it also makes any combination of them zero; so all those planes contain $r$.
If the line is given in symmetric or parametric form, first write it as the intersection of two planes (setting the fractions equal two at a time). Other conditions translate as follows: perpendicular to a plane $\sigma$ means $\vec n\cdot\vec n_\sigma=0$ (this gives an equation for $\lambda$); parallel to a line means $\vec n\cdot\vec d=0$.
Pencil of Planes and the Tetrahedron of Minimum Volume
If you ask for the plane in the pencil that forms a tetrahedron of minimum volume with the axes, you combine the pencil, the intercept equation, and the derivative (optimization: model, differentiate, and justify the minimum).
Consider the pencil of planes whose axis is the line of intersection of the planes $\pi_1 \equiv 2x - y + z - 3 = 0$ and $\pi_2 \equiv x + y - 2z + 1 = 0$. Find the…
Consider the pencil of planes whose axis is the line of intersection of the planes $\pi_1 \equiv 2x - y + z - 3 = 0$ and $\pi_2 \equiv x + y - 2z + 1 = 0$. Find the plane belonging to this pencil that passes through the midpoint of the line segment with endpoints $A(1, 2, -1)$ and $B(3, 0, 5)$.
Consider the pencil of planes whose axis is the line $r: \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z}{3}$. Calculate the equation of the plane $\pi$ belonging to this…
Consider the pencil of planes whose axis is the line $r: \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z}{3}$. Calculate the equation of the plane $\pi$ belonging to this pencil that satisfies the following two conditions simultaneously:
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It is perpendicular to the plane $\sigma: x - 2y + z - 5 = 0$.
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The point $P(0, a, 1)$ belongs to the plane $\pi$ for some real value of $a$.
Determine the value of $a$ and the general equation of $\pi$.
Find the equation of the plane $\pi$ that contains the line of intersection of the planes $\alpha: 2x - y + 3z = 1$ and $\beta: x + 2y - z = 0$, and that intersects the…
Find the equation of the plane $\pi$ that contains the line of intersection of the planes $\alpha: 2x - y + 3z = 1$ and $\beta: x + 2y - z = 0$, and that intersects the positive coordinate semi-axes $OX$, $OY$, and $OZ$ at points $P$, $Q$, and $R$ such that the volume of the tetrahedron formed by the origin and these points is minimized.
Distance, Angle, and Relative Position of Planes
Several problems about planes need three quick tools (they are studied in depth in other sections).
Why? The distance is the length of the projection of $\vec{QP}$ onto $\vec n$, with $Q$ a point of the plane; the angle between planes is that of their perpendiculars.
Plane parallel to a line. A line with direction $\vec d$ is parallel to the plane if $\vec n\cdot\vec d=0$; then all its points are at the same distance from the plane, and that is the line–plane distance. To require that a plane be parallel to two lines, take $\vec n=\vec d_r\times\vec d_s$.
Three planes. To study their relative position depending on a parameter $k$, set up the $3\times3$ system and solve it by the elimination method (Gaussian elimination), discussing when it is consistent with a unique solution, with infinitely many solutions, or inconsistent.
Given the planes $\pi_1 \equiv (a-2)x + 2y + z = 0$ and $\pi_2 \equiv 4x + ay + 2z = b$: a) Calculate the values of $a$ and $b$ so that the planes are coincident. b)…
Given the planes $\pi_1 \equiv (a-2)x + 2y + z = 0$ and $\pi_2 \equiv 4x + ay + 2z = b$:
- a) Calculate the values of $a$ and $b$ so that the planes are coincident.
- b) Calculate the values of $a$ and $b$ so that the planes are parallel but not coincident.
- c) What relationship must exist between $a$ and $b$ so that the origin of coordinates belongs to the intersection of both planes?
Consider the intersecting lines $r$ and $s$ defining a plane $\alpha$. Line $r$ passes through $P(1, 0, 1)$ with direction $\vec{d_r} = (2, 1, -1)$ and line $s$ passes…
Consider the intersecting lines $r$ and $s$ defining a plane $\alpha$. Line $r$ passes through $P(1, 0, 1)$ with direction $\vec{d_r} = (2, 1, -1)$ and line $s$ passes through $Q(1, 0, 1)$ with direction $\vec{d_s} = (0, 1, 1)$.
Calculate the equation of a plane $\pi$ parallel to $\alpha$ such that the sphere with center $C(3, 2, 1)$ and radius $R=2$ is tangent to this plane $\pi$. Obtain the two possible solutions.
A drone moves in a straight line following the path of line $r: \frac{x-1}{2} = \frac{y-1}{1} = \frac{z-1}{2}$. It is desired to install a flat protective screen $\pi$…
A drone moves in a straight line following the path of line $r: \frac{x-1}{2} = \frac{y-1}{1} = \frac{z-1}{2}$. It is desired to install a flat protective screen $\pi$ containing line $s: \begin{cases} x = t \\ y = 2 \\ z = 1-t \end{cases}$.
Find the equation of plane $\pi$ such that it is parallel to the drone's path. Calculate the minimum distance at which the drone will pass from this screen.