Equation of Straight Line in Space

Theorem

Let $\LL$ be a straight line embedded in a cartesian $3$-space passing through the point $\tuple {x_1, y_1, z_1}$.

$\LL$ is expressed in symmetric form by the equation:

$\dfrac {x - x_1} l = \dfrac {y - y_1} m = \dfrac {z - z_1} n$

where $l, m, n \in \R$ are the direction ratios of $\LL$.

Let $\LL$ be a straight line embedded in a cartesian $3$-space passing through the points $\tuple {x_1, y_1, z_1}$ and $\tuple {x_2, y_2, z_2}$.

$\LL$ is expressed in two-point form by the equation:

$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1} = \dfrac {z - z_1} {z_2 - z_1}$

Let $\LL$ be a straight line embedded in a cartesian $3$-space passing through the point $\tuple {x_1, y_1, z_1}$ and with direction cosines $l$, $m$ and $n$.

$\LL$ is expressed in parametric form by the set of equations:

$\begin {cases} x & = & x_1 + l d \\ y & = & y_1 + m d \\ z & = & z_1 + n d \end {cases}$

where $d \in \R$ is the distance of the variable point $\tuple {x, y, z}$ from $\tuple {x_1, y_1, z_1}$

Let $\mathbf a$ and $\mathbf b$ denote the position vectors of two points in space

Let $L$ be a straight line in space passing through $\mathbf a$ which is parallel to $\mathbf b$.

Let $\mathbf r$ be the position vector of an arbitrary point on $L$.

Then:

$\mathbf r = \mathbf a + t \mathbf b$

for some real number $t$, which may be positive or negative, or even $0$ if $\mathbf r = \mathbf a$.