Equation of Straight Line in Space/Parametric Form

Theorem

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 $\tuple {x_1, y_1, z_1}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): line: 2.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parametric equations
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parametric equations