# Parallel Lines in space

1. May 16, 2015

### Cpt Qwark

1. The problem statement, all variables and given/known data
How are the two lines
r = i + 2j + t(i - k), and r = k + s(-i + k)
parallel?
t,s∈ℝ

2. Relevant equations
parametric vector equation of a line
$$r-r_0=tv$$
3. The attempt at a solution
Tried to find the conditions for lines to be parallel in ℝ^3.

2. May 16, 2015

### haruspex

Suppose you have the equation as a constant vector $\vec a$ plus some parameter mutiplied by a second constant vector $\vec b$. What scalar and vector operations can you do to it that would produce parallel lines?

3. May 16, 2015

### HallsofIvy

Staff Emeritus
A line in space can be written as $\vec{r}= \vec{r_0}+ \vec{D}t$ where $\vec{r_0}$ is the "position vector" of a single point on the line (the point where t= 0) and $\vec{D}$ is the "direction vector" pointing in the direction of the line. Two lines are parallel if and only if one direction vector is a multiple of the other.

Edit: Some text removed by a mentor.

Last edited by a moderator: May 16, 2015