Equations of Lines and Planes

Definition Let $P_0$ be a point with position vector $\mathbf{r}_0 = \langle x_0, y_0, z_0 \rangle$ and let $\mathbf{v} = \langle a, b, c \rangle$ be a nonzero vector. The line through $P_0$ parallel to $\mathbf{v}$ is the set of points with position vector

In components this is the parametric form

The vector $\mathbf{v}$ is a direction vector. Any nonzero multiple of $\mathbf{v}$ and any other point of the line describe the same line, so a line has many equations.

Definition If $a$, $b$ and $c$ are all nonzero, solving each parametric equation for $t$ gives the symmetric equations

If one component is zero, that fraction is not allowed. For example, if $b = 0$ while $a, c \ne 0$, the line is

If two components are zero, say $a = b = 0$, the line is $x = x_0$, $y = y_0$ with $z$ arbitrary.

Definition Let $P_0$ have position vector $\mathbf{r}_0$ and let $\mathbf{n} = \langle a, b, c \rangle$ be a nonzero vector. The plane through $P_0$ with normal vector $\mathbf{n}$ is the set of points with position vector $\mathbf{r}$ such that $\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0$, that is

This can be written $ax + by + cz = d$ with $d = \mathbf{n} \cdot \mathbf{r}_0$. Conversely, every equation $ax + by + cz = d$ with $a, b, c$ not all zero is a plane with normal $\langle a, b, c \rangle$. Through three points $P, Q, R$ that are not on one line there is exactly one plane, and $\mathbf{n} = \overrightarrow{PQ} \times \overrightarrow{PR}$ is a normal to it.

Definition The angle between two lines with direction vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ is the angle $\theta$ in $[0, \frac{\pi}{2}]$ with

The angle between two planes is the angle between their normals, given by the same formula with $\mathbf{n}_1, \mathbf{n}_2$. The planes are parallel or equal exactly when their normals are parallel. The angle between a line with direction $\mathbf{v}$ and a plane with normal $\mathbf{n}$ is $\varphi = \frac{\pi}{2} - \alpha$, where $\alpha$ in $[0, \frac{\pi}{2}]$ is the angle between the line and the normal line. So $\varphi$ lies in $[0, \frac{\pi}{2}]$, it is $0$ when the line is parallel to the plane and $\frac{\pi}{2}$ when the line is perpendicular to it, and

Definition To intersect the line $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$ with the plane $\mathbf{n} \cdot \mathbf{r} = d$, substitute to get $\mathbf{n} \cdot \mathbf{r}_0 + t\,(\mathbf{n} \cdot \mathbf{v}) = d$. If $\mathbf{n} \cdot \mathbf{v} \ne 0$ there is exactly one intersection point, at

If $\mathbf{n} \cdot \mathbf{v} = 0$ the line is parallel to the plane. It lies in the plane when $\mathbf{n} \cdot \mathbf{r}_0 = d$ and misses it otherwise. Two planes with nonparallel normals meet in a line with direction $\mathbf{n}_1 \times \mathbf{n}_2$. Two planes with parallel normals are either the same plane or do not meet.

Definition Two lines in space are skew if they are not parallel and do not intersect. For the lines $\mathbf{r}_1 + s\mathbf{v}_1$ and $\mathbf{r}_2 + t\mathbf{v}_2$ this happens exactly when

Their distance, the length of the shortest segment joining them, is

When the cross product is nonzero but the triple product is zero, the lines lie in one plane and intersect. Skew lines do not exist in the plane, where nonparallel lines always meet.

Definition The distance from the point $P_1(x_1, y_1, z_1)$ to the plane $ax + by + cz = d$, where $a, b, c$ are not all zero, is

It is the length of the projection of $\overrightarrow{P_0 P_1}$ onto the normal, for any point $P_0$ of the plane. The distance between parallel planes $ax + by + cz = d_1$ and $ax + by + cz = d_2$ is $\lvert d_1 - d_2 \rvert / \sqrt{a^2 + b^2 + c^2}$.