Essential Concepts

• In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector ${\bf{v}}=\langle{a,b,c}\rangle$ passing through point $P=(x_{0},y_{0},z_{0})$ is ${\bf{r}}={\bf{r}}_{0}+t{\bf{v}}$, where ${\bf{r}}_{0}=\langle{x_{0},y_{0},z_{0}}\rangle$ is the position vector of point $P$ This equation can be rewritten to form the parametric equations of the line: $x=x_{0}+ta$, $y=y_{0}+tb$, and $z=z_{0}+tc$.  The line can also be described with the symmetric equations $\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}$.
• Let $L$ be a line in space passing through point $P$ with direction vector ${\bf{v}}$. If $Q$ is any point not on $L$ then the distance from $Q$ to $L$ is $d=\frac{\parallel\overrightarrow{PQ}\times{\bf{v}}\parallel}{\parallel{\bf{v}}\parallel}$.
• In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
• Given a point $P$ and vector ${\bf{n}}$ the set of all points $Q$ satisfying equation ${\bf{n}}\cdot\overrightarrow{PQ}=0$ forms a plane. Equation ${\bf{n}}\cdot\overrightarrow{PQ}=0$ is known as the vector equation of a plane.
• The scalar equation of a plane containing point $P=(x_{0},y_{0},z_{0})$ with normal vector ${\bf{n}}=\langle{a,b,c}\rangle$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$. This equation can be expressed as $ax+by+cz+d=0$, where $d=-ax_{0}-by_{0}-cz_{0}$. This form of the equation is sometimes called the general form of the equation of a plane.
• Suppose a plane with normal vector ${\bf{n}}$ passes through point $Q$. The distance $D$ from the plane to point $P$ not in the plane is given by $D=\parallel\text{proj}_{\bf{n}}\overrightarrow{QP}\parallel=|\text{comp}_{\bf{n}}\overrightarrow{QP}|=\dfrac{|\overrightarrow{QP}\cdot{\bf{n}}|}{\parallel{\bf{n}}\parallel}$.
• The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
• The measure of the angle $\theta$ between two intersecting planes can be found using the equation: $\cos\theta=\frac{|{\bf{n}}_{1}\cdot{\bf{n}}_{2}|}{\parallel{\bf{n}}_{1}\parallel\parallel{\bf{n}}_{2}\parallel}$, where ${\bf{n}}_{1}$ and ${\bf{n}}_{2}$ are normal vectors to the planes.
• The distance $D$ from the point $(x_{0},y_{0},z_{0})$ to plane $ax+by+cz+d=0$ is given by $D=\dfrac{|a(x_{0}-x_{1})+b(y_{0}-y_{1})+c(z_{0}-z_{1})|}{\sqrt{a^2+b^2+c^2}}=\dfrac{|ax_{0}+by_{0}+cz_{0}+d|}{\sqrt{a^2+b^2+c^2}}$