Question

Derive the equation of the line in space passing through a point and parallel to a vector both in vector and Cartesian form.

Answer 1

A line is determined by a point and a direction. Thus, to find an equation representing a line in three dimensions choose a point $P_0$ on the line and a non-zero vector ${}^{\prime }{v}^{\prime }$ parallel to the line.

Since any constant multiple of a vector still points in the same direction, it seems reasonable that a point on the line can be found by starting at the point $P_0$ on the line and following a constant multiple of the vector v (see the figure above).

If ${}^{\prime }{r}^{\prime }$ is the position vector of $P$, then the line must be in the form

$\overrightarrow{r}={\overrightarrow{r}}_0+t\overrightarrow{v}$

Let the direction ratios of the line be $a,b,c$. Consider coordinates of any point $A$ be $(x,y,z)$.

Then Cartesian form of a line is given by

$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$