TITLE: What is the Equation for a straight line in a 3D space? And how to find other parelell lines to it?
QUESTION [7 upvotes]: We know $y=mx+c$ is the equation for a straight line in a 2D graph. And the parallel line that goes through $(x_1,y_1)$  is $y=mx+(-mx_1+y_1)$. But how do we display the straight line in a 3D graph from an equation? How does we identify the line which is parallel in all 3 axis that goes through point $(x_1,y_1,z_1)$?

REPLY [3 votes]: Using vector notation it is straightforward to represent a line, in any number of dimensions. 
$2-$dimensional: $(x, y) = (5, 3) + m  (1, 0)$
$3-$dimensional: $(x, y, z) = (5, 3, 1) + m  (1, 0, 0)$