Equation of a Line in Space (Cartesian and Vector Form)

A straight line in space is completely determined by (a) one point on it and its direction, or (b) any two distinct points on it (which together fix both a point and, via their difference, a direction) -- so its equation can always be built from these two ingredients, in either vector or Cartesian form.

Vector equation through a point, parallel to a given vector. Let the line pass through a point $A$ with position vector $\vec a$, and be parallel to a given vector $\vec b=a\hat i+b\hat j+c\hat k$ (not the zero vector). Let $P$ be any point on the line, with position vector $\vec r$. Since $P$ lies on the line through $A$ in the direction $\vec b$, the vector $\vec{AP}=\vec r-\vec a$ must be parallel to $\vec b$, i.e. equal to some scalar multiple of it: $\vec r-\vec a=\lambda\vec b$ for some real number $\lambda$ (a parameter). Rearranging gives the vector equation of a line:

As $\lambda$ ranges over all real numbers, $\vec r$ traces out every point of the line exactly once; $\lambda=0$ gives the point $A$ itself.

Cartesian (symmetric) form -- derivation. Write $\vec r=x\hat i+y\hat j+z\hat k$ for a general point $P(x,y,z)$ on the line, and $\vec a=x_1\hat i+y_1\hat j+z_1\hat k$ for the known point $A(x_1,y_1,z_1)$. Substituting into $\vec r=\vec a+\lambda\vec b$ and equating the $\hat i$, $\hat j$, $\hat k$ components separately gives three scalar equations,

Solving each for $\lambda$ (whenever $a,b,c\ne0$) gives $\lambda=\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$, i.e. the Cartesian equation of a line in symmetric form:

Here $(x_1,y_1,z_1)$ is any known point on the line and $a,b,c$ are its direction ratios (any proportional triple works equally well, since a common scalar factor cancels from all three fractions). If one of $a,b,c$ is zero, the corresponding coordinate is simply constant along the line (e.g. $a=0$ means every point has $x=x_1$); the symmetric form is still written with that zero in the denominator, understood as this convention rather than an actual division. …