Equation of a Line in Space

Equation of a Line in Space

In two-dimensional geometry, a line is determined by a point and a slope, or by two points. In three-dimensional space, a line is uniquely determined if we know either:

• (i) a point through which it passes and its direction, or
• (ii) two distinct points through which it passes.

We develop both the vector and cartesian forms of the equation of a line in space.

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Equation of a Line through a Given Point and Parallel to a Given Vector

Let a line pass through a point $A$ with position vector $\vec{a}$, parallel to a given vector $\vec{b}$. For any point $P$ on the line with position vector $\vec{r}$, the vector $\overrightarrow{AP} = \vec{r} - \vec{a}$ is parallel to $\vec{b}$, so there is a scalar $\lambda$ with $\vec{r} - \vec{a} = \lambda \vec{b}$. This gives the vector equation:

As $\lambda$ takes all real values, $P$ traces the entire line. The vector $\vec{b}$ is the direction vector of the line.

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Cartesian Form

Let $A = (x_1, y_1, z_1)$, direction ratios $a, b, c$ (so $\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}$), and $P = (x, y, z)$. Substituting into $\vec{r} = \vec{a} + \lambda \vec{b}$ and equating the coefficients of $\hat{i}, \hat{j}, \hat{k}$ gives the parametric equations:

Eliminating the parameter by solving each for $\lambda$:

Since $\lambda$ is common to all three, we obtain the cartesian (symmetric) equation:

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Equation of a Line through Two Given Points

Let the line pass through $A$ and $B$ with position vectors $\vec{a}$ and $\vec{b}$. Its direction vector is $\overrightarrow{AB} = \vec{b} - \vec{a}$, so the vector equation is:

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Cartesian Form …