Finding the Equation of a Line

Vector and Cartesian form

A line through a point with position vector $\vec{a}$, parallel to direction vector $\vec{b}$, has vector equation $\vec{r} = \vec{a} + \lambda\vec{b}$. If $\vec{a}=(x_1,y_1,z_1)$ and $\vec{b}=(a,b,c)$, the same line in Cartesian form is:

Worked example: through two points

Find the equation of the line through $(3,-2,-5)$ and $(3,-2,6)$.

Solution: The direction ratios are the difference of the two points: $(3-3, -2-(-2), 6-(-5)) = (0,0,11)$.

The zeroes in the denominator just mean $x$ and $y$ stay fixed at 3 and -2 while $z$ varies, matching a line running straight along the $z$-direction.

Worked example: point and parallel vector

Find the vector equation of the line through $2\hat{i}+3\hat{j}+\hat{k}$ and parallel to $4\hat{i}-2\hat{j}+3\hat{k}$.

Solution: Plug directly into $\vec{r}=\vec{a}+\lambda\vec{b}$: