# Distance between two parallel lines in 3-space [closed]

I know these two lines are parallel, but I don't know how to find the distance between them. Any suggestions?

Line 1: (3, 4, 7) + {6, 2, 4}t

Line 2: (-1, 5, -1) + {3, 1, 2}t

I tried creating a triangle composed of an orthogonal line to both lines (the shortest distance between the lines), a hypotenuse that connects the two known points (3, 4, 7) and (-1, 5, -1), and a leg that connects (-1, 5, -1) to the intersection of the orthogonal line on line 2. On line 1, the point (3, 4, 7) was the intersection of the orthogonal line and the hypotenuse.

I'm sorry if that doesn't make any sense, I did my best. I have no idea how to include pictures on this website, this is my first post. And this is not a homework assignment, it's a practice problem for a Multivariable Calculus class. I'm studying for a test we have next week. Any and all help is appreciated!

• This is off-topic here. Perhaps try asking on math.SE. – Ian MacDonald May 26 '16 at 17:04
• Hint: the component of a vector $\vec{u}$ perpendicular to $\vec{v}$ is $\vec{u}-\vec{v}\frac{\vec{u}\cdot\vec{v}}{\vec{v}\cdot\vec{v}}$. – f'' May 26 '16 at 17:17