Lines in Space

The Parametric Equations of a Line

The Geometric Meaning of the Parameter

The parameter $t$ represents the “step size” of moving from the starting point $P_0$ in the direction $\vec{s}$:

• $t > 0$: moving in the positive direction
• $t < 0$: moving in the opposite direction
• $t = 0$: at the starting point

The Symmetric Equations of a Line

The Features of the Symmetric Form

• the form is concise and pleasingly symmetric
• the direction vector appears directly in the denominators
• it is suitable for computing the coordinates of points on the line

The Two-Point Form of a Line

Applications of the Two-Point Form

• when two points on the line are known, use the two-point form directly
• the direction vector is $\vec{P_1P_2} = (x_2-x_1, y_2-y_1, z_2-z_1)$

The General Form of a Line

The Features of the General Form

• it is expressed as the intersection of two planes
• it is suitable for the case where the line intersects planes
• it does not give the direction vector directly

Converting between the Forms

The Parametric Equations → the Symmetric Equations

From the parametric equations: $\begin{cases} x = x_0 + lt \\ y = y_0 + mt \\ z = z_0 + nt \end{cases}$

convert to the symmetric form: $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$

The Symmetric Equations → the Parametric Equations

From the symmetric form: $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n} = t$

convert to the parametric equations: $\begin{cases} x = x_0 + lt \\ y = y_0 + mt \\ z = z_0 + nt \end{cases}$

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Exercises

Exercise 1

Find the equation of the line passing through the point $A(1, 2, 3)$ with direction vector $(1, -1, 2)$.

Exercise 2

Find the equation of the line passing through the two points $A(1, 2, 3)$ and $B(2, 3, 4)$.

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Summary

Symbols Used in This Article