I have been given a definition in my geometry notes that two lines in $\Bbb R^3$ are parallel if
$1)$ They lie in the same plane
$2)$ They do not intersect.
How can I use this to show that they have the same direction vector. Suppose they lie in the plane $H = \{ v \in \Bbb R^3 | a \cdot v = b \}$ for some $b \in \Bbb R^3$. Then let the lines be named $L_1, L_2$. Not sure how to proceed. Hints appreciated.
Best Answer
Changing coordinates, you may as well assume that $L_1$ and $L_2$ lie in the $xy$-plane and that neither passes through the origin. In this case the equation for $L_i$ is $a_i x + b_i y = 1$ for non-zero $a_i$ and $b_i$. Assuming that $L_1$ and $L_2$ are not the same line, you get a common solution to these equations unless $(a_1,b_1)$ is a scalar multiple of $(a_2,b_2)$. Since these are the normal vectors to $L_1$ and $L_2$, you get the result you want.