Planes

Vector equation

A plane is a flat surface in three-dimensional space that extends indefinitely in all directions. The vector equation of a plane containing point with position vector and parallel to the directions of vectors and is:

Conceptually, this is very similar to the vector equation of a line, first going to the plane, then travelling along it parallel to the two direction vectors.

A plane is uniquely defined by three non-colinear points. Alternatively, a plane can also be defined by

a line and a point not on the line
two intersecting lines with direction vectors parallel to the plane.

Cartesian equation

The Cartesian equation of a plane is often more useful, as it is difficult to tell if two planes in vector equation form are identical or parallel. The Cartesian equation uses the normal vector, a vector perpendicular to every line in the plane.

Let be a point in the plane with position vector , and be any other point in the plane. The vector has to be perpendicular to the normal vector , giving us the Cartesian equation:

By rewriting and , we get:

To convert from vector form to Cartesian form, use the cross product of the two direction vectors to find the normal vector of the plane. Then, use any point in the plane to find the constant dot product.

To convert from Cartesian form to vector form, find any three points in the plane and then use these points to find two direction vectors. For simplicity, try substituting different combinations of .

Intersections between lines and planes

The following method requires the equation of the plane in Cartesian form and the equation of the line in vector form:

Using the vector equation of the line, find expressions in terms of the parameter (often ) for the components.
Substitute each into the Cartesian plane equation.
Solve for (or if the equation is inconsistent, the line and plane do not intersect).

Angles between lines and planes

The angle between the line with direction vector and the plane with normal is , where is the acute angle between and .

The angle between two planes is the angle between their normals.

Distances between points, lines, and planes

Distance between a point and plane
For a plane with equation , and a point outside of the plane, the shortest distance from to the plane is where is perpendicular to the plane, making a foot of the perpendicular.

To find the distance :

Find the vector equation of the line with direction vector through .
Find the intersection point between this line and the plane.
Find the distance .

Following this gives the formula:

Where is the point being considered.

Distance between a point and line
The shortest distance between a point and the line with equation is given by:

This formula is given.

Distance between two skew lines
Consider two points and along two skew lines. The distance is minimised when the line segment is perpendicular to both lines 1.

The shortest distance between two skew lines with equations and is:

where .

Distance between two parallel lines
Consider two parallel lines with direction vectors . For a fixed point on the first line and a point on the second line, the distance is minimised when is perpendicular to both lines. The distance is equal to the distance:

D = |\boldsymbol{a} - \boldsymbol{b}|\sin\theta $$Where $\theta$ is the angle between $AP$ and $d$, so $\cos\theta = \frac{(\boldsymbol{a} - \boldsymbol{b})\cdot\boldsymbol{d}}{|\boldsymbol{a} - \boldsymbol{b}||\boldsymbol{d}|}$. *This is not given.* ## Simultaneous equations and planes #### Two variables For **simultaneous linear equations in two variables**, which can be written as $\boldsymbol{M}\boldsymbol{X}=\boldsymbol{B}$, there are three possible cases: - One unique solution, when $|\boldsymbol{M}| \not= 0$ (two intersecting lines). - Infinitely many solutions, when $|\boldsymbol{M}| = 0$ and the equations are consistent (two overlapping parallel lines). - No solutions, when $|\boldsymbol{M}| = 0$ and the equations are not consistent (two non-overlapping parallel lines). A visual representation is provided below. ```tikz \usepackage{amsmath} \begin{document} \begin{tikzpicture}[>=stealth, scale=0.8] \begin{scope}[shift={(0,0)}] \draw[->] (-0.5,0) -- (4,0) node[right] {$x$}; \draw[->] (0,-0.5) -- (0,4) node[above] {$y$}; \draw[thick, blue] (-0.5, 0.5) -- (3.5, 3.5) node[right] {$L_1$}; \draw[thick, red] (-0.5, 3.5) -- (3.5, 0.5) node[right] {$L_2$}; \filldraw[black] (1.5,1.5) circle (2pt); \node[align=center, below] at (1.5, -1) { \textbf{Unique solution} \\ $|\mathbf{M}| \neq 0$ \\ Lines intersect }; \end{scope} \begin{scope}[shift={(6,0)}] \draw[->] (-0.5,0) -- (4,0) node[right] {$x$}; \draw[->] (0,-0.5) -- (0,4) node[above] {$y$}; \draw[ultra thick, blue] (-0.5, 1) -- (3.5, 3) node[right] {$L_1, L_2$}; \draw[dashed, white, thick] (-0.5, 1) -- (3.5, 3); \node[align=center, below] at (1.5, -1) { \textbf{Infinite solutions} \\ $|\mathbf{M}| = 0$ (consistent) \\ Lines overlap }; \end{scope} \begin{scope}[shift={(12,0)}] \draw[->] (-0.5,0) -- (4,0) node[right] {$x$}; \draw[->] (0,-0.5) -- (0,4) node[above] {$y$}; \draw[thick, blue] (-0.5, 2) -- (3.5, 4) node[right] {$L_1$}; \draw[thick, red] (-0.5, 0.5) -- (3.5, 2.5) node[right] {$L_2$}; \node[align=center, below] at (1.5, -1) { \textbf{No solution} \\ $|\mathbf{M}| = 0$ (inconsistent) \\ Lines are parallel }; \end{scope} \end{tikzpicture} \end{document} ``` #### Three variables For **simultaneous linear equations in three variables**, there are six possible cases: - One unique solution, when $|\boldsymbol{M}| \not= 0$ where three planes intersect at a point. - Infinitely many solutions, when $|\boldsymbol{M}| = 0$ and the equations are consistent. - Three planes intersect at a line, forming a sheaf. - Three overlapping planes. - No solutions, when $|\boldsymbol{M}| = 0$ and the equations are not consistent. - When three planes are all parallel to each other - When two planes are parallel and one is not - When the planes enclose a triangular prism, with all pairs of planes intersecting at parallel lines. Two planes are parallel if their normal vectors are parallel (are multiples of each other). **Footnotes** [^1]: **Footnote on skew lines and perpendicularity** It is not apparent that such a line segment necessarily exists, but it does for all pairs of skew lines.