Equation Of Line

Question

The symmetric form of equation of a line passing through (x₁, y₁, z₁) with direction ratios a, b, c is

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1. **Vector Theory**: A line passing through point $\vec{A} = (x_1, y_1, z_1)$ and parallel to a direction vector $\vec{D} = a\hat{i} + b\hat{j} + c\hat{k}$ is given by $\vec{r} = \vec{A} + \lambda\vec{D}$.
2. **Cartesian Translation**: Separating the components:
   - $x = x_1 + \lambda a \implies \lambda = \frac{x-x_1}{a}$
   - $y = y_1 + \lambda b \implies \lambda = \frac{y-y_1}{b}$
   - $z = z_1 + \lambda c \implies \lambda = \frac{z-z_1}{c}$
3. **Symmetric Form**: Equating the $\lambda$ values: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$.

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Equation Of Line

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