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Analytic geometry of three dimensions: (lesson 1 of 2)

## Line in three dimensions

### 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:

$$\left| {\begin{array}{*{20}{c}} {x_1 - x_0}&{y_1 - y_0}&{z_1 - z_0}\\ {a_0}&{b_0}&{c_0}\\ {a_1}&{b_1}&{c_1} \end{array}} \right| = 0$$

### 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

$$\left| {\begin{array}{*{20}{c}} {a_0}&{a_1}&{a_2}\\ {b_0}&{b_1}&{b_2}\\ {c_0}&{c_1}&{c_2} \end{array}} \right| = 0$$

### Distance:

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

$$D = \sqrt {\frac{{{{\left| {\begin{array}{*{20}{c}} {y_0 - y_1}&{z_o - z_1}\\ b&c \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} {z_0 - z_1}&{x_o - x_1}\\ c&a \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} {x_0 - x_1}&{y_o - y_1}\\ a&b \end{array}} \right|}^2}}}{{{a^2} + {b^2} + {c^2}}}}$$

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 $(a_1, b_1, c_1)$:

$$D = \frac{{\left| {\begin{array}{*{20}{c}} {{x_1} - {x_0}}&{{y_1} - {y_0}}&{{z_1} - {z_0}}\\ {{a_0}}&{{b_0}}&{{c_0}}\\ {{a_1}}&{{b_1}}&{{c_1}} \end{array}} \right|}}{{\sqrt {{{\left| {\begin{array}{*{20}{c}} {{b_0}}&{{c_o}}\\ {{b_1}}&{{c_1}} \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} {{c_0}}&{{a_o}}\\ {{c_1}}&{{a_1}} \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} {{a_0}}&{{b_0}}\\ {{a_1}}&{{b_1}} \end{array}} \right|}^2}} }}$$