Cartesian Equation of a Line

Line through (x₁, y₁, z₁) with direction ratios (a, b, c). Each ratio equals the parameter λ. If any direction ratio is zero (say a = 0), the corresponding equation is x = x₁.

The line through $(x_1, y_1, z_1)$ with DRs $(a, b, c)$ has parametric equations:

$$x = x_1+\lambda a, \quad y = y_1+\lambda b, \quad z = z_1+\lambda c$$

Solving each for $\lambda$:

$$\lambda = \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$

This is the symmetric (Cartesian) form. Each ratio equals $\lambda$ — the common parameter.

When a direction ratio is zero: If $a = 0$, the line is perpendicular to the x-axis, so $x = x_1$ throughout. Write: $x = x_1$, $(y-y_1)/b = (z-z_1)/c$.

Reading DRs from the equation: Given $(x-2)/3 = (y+1)/(-4) = z/5$, the point is $(2,-1,0)$ and DRs are $(3,-4,5)$.

Converting between forms:

• Vector → Cartesian: read $\mathbf{a}$ as the point and $\mathbf{b}$ as the DRs
• Cartesian → Vector: $\mathbf{r} = (x_1\hat{i}+y_1\hat{j}+z_1\hat{k}) + \lambda(a\hat{i}+b\hat{j}+c\hat{k})$