Planes »

Equations:

Symmetric form for describing the straight line:

1. Line through $(x_0, y_0, z_0)$ parallel to the vector $(a, b, c)$:

$\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$

2. Line through point $(x_0, y_0, z_0)$ and $(x_1, y_1, z_0)$:

$\frac{x - x_0}{x_1 - x_0} = \frac{y - y_0}{y_1 - y_0} = \frac{z - z_0}{z_1 - z_0}$

This line is parallel to the vector $(x_1 - x_0, y_1 - y_0, z_1 - z_0)$

Parametric Form

In three-dimensional space, the line passing through the point $(x_0, y_0, z_0)$ and is parallel to $(a, b, c)$ has parametric equations

$$\begin{aligned} x &= x_0 + at \\ y &= y_0 + bt \\ z &= z_0 + ct \\ -\infty &< t < + \infty \end{aligned}$$

In three-dimensional space, the line passing through the points $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$

$$\begin{aligned} x &= x_0 + (x_1 - x_0) t \\ y &= y_0 + (y_1 - y_0) t \\ z &= z_0 + (z_1 - z_0) t \\ - \infty &< t < + \infty \end{aligned}$$

Intersection:

The line through $(x_0, y_0, z_0)$ in direction $(a_0, b_0, c_0)$, and the line through $(x_1, y_1, z_1)$ in direction $(a_1, b_1, c_1)$, intersect if:

\left| {} \right| = 0

Parallelism:

Three lines with directions $(a_0, b_0, c_0)$, $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel to a common plane if and only if

\left| {} \right| = 0

Distance:

Distance between the point $(x_0, y_0, z_0)$ and the line through $(x_1, y_1, z_1)$ in direction $(a, b, c)$:

D = \sqrt {\frac{{{{\left| {} \right|}^2} + {{\left| {} \right|}^2} + {{\left| {} \right|}^2}}}{{{a^2} + {b^2} + {c^2}}}}

Distance between the line through $(x_0, y_0, z_0)$, in direction $(a_0, b_0, c_0)$, and the line through $(x_1, y_1, z_1)$, in direction $(a_1, b_1, c_1)$:

D = \frac{{\left| {} \right|}}{{\sqrt {{{\left| {} \right|}^2} + {{\left| {} \right|}^2} + {{\left| {} \right|}^2}} }}