Line Through Two Points

Line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Direction ratios are (x₂−x₁, y₂−y₁, z₂−z₁).

For $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the direction vector of the line is:

$$\mathbf{b} = \overrightarrow{P_1P_2} = (x_2-x_1,\; y_2-y_1,\; z_2-z_1)$$

Line through $P_1$ with this direction:

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

Parameter meaning: $\lambda = 0$ gives $P_1$; $\lambda = 1$ gives $P_2$; $\lambda = 1/2$ gives the midpoint.

Collinearity of three points $A$, $B$, $C$: $C$ lies on line $AB$ iff the DRs of $AB$ and $AC$ are proportional:

$$\frac{x_C-x_A}{x_B-x_A} = \frac{y_C-y_A}{y_B-y_A} = \frac{z_C-z_A}{z_B-z_A}$$

Alternatively, $\overrightarrow{AB} \times \overrightarrow{AC} = \mathbf{0}$.