Name

Form

Details

Vector

$\mathbf{R}\=\mathbf{P}\+t \mathbf{V}$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]\+t \left[\begin{array}{c}u\\ v\\ w\end{array}\right]$

Parametric

$x\=x\left(t\right)\,y\=y\left(t\right)\,z\=z\left(t\right)$

$x\=a\+t u\, y\=b\+t v\,z\=c\+t w$

Symmetric

$\frac{x-a}{u}\=\frac{y-b}{v}\=\frac{z-c}{w}$

Table 1.6.1   Forms for a line in ${\mathrm{\ℝ}}^3$ 

Example 1.6.1

Obtain an equation for the line that passes through the point P:$\left(3\,2\,1\right)$ and that is parallel to the vector $\mathbf{V}\=2 \mathbf{i}-3 \mathbf{j}\+5 \mathbf{k}$.

Example 1.6.2

Obtain an equation for the line through the points P:$\left(3\,2\,1\right)$ and Q:$\left(5\,-1\,4\right)$.

Example 1.6.3

The lines $\mathbf{R}\=\mathbf{A}\+t \mathbf{P}$ and $\mathbf{R}\=\mathbf{B}\+s \mathbf{Q}$, are defined respectively by the parametric equations

$x\=3-2 t\,y\=2\+5 t\,z\=6\+t$  and  $x\=5\+4 s\,y\=7\+2 s\,z\=3\+2 s$

Example 1.6.4

Find the distance from the point P:$\left(2\,1\,-3\right)$ to the line $\mathbf{R}\=\mathbf{A}\+t \mathbf{V}$, where $\mathbf{A}\=\mathbf{i}-\mathbf{j}\mathbf{\+}2 \mathbf{k}$ and $\mathbf{V}\=7 \mathbf{i}-9 \mathbf{j}\+5 \mathbf{k}$.

Example 1.6.5

Lines $L_1$ and $L_2$ both have the common direction $\mathbf{V}\=3 \mathbf{i}-2 \mathbf{j}\+5 \mathbf{k}$, with $L_1$ passing through point P:$\left(7\,-4\,6\right)$ and $L_2$ passing through Q:$\left(4\,-7\,1\right)$.