# Two planes contain two non intersecting lines can be made parallel?

1. Mar 31, 2013

### yungman

From the book, given any two lines non intersecting lines in 3 space, any two planes each contains one of the lines can always make parallel to each other ( the two planes are parallel).

The way the book described is that given two lines, you can produce two vectors V1 and V2, each parallel to one of the lines. Then by cross product V1 X V2, you get the normal vector N. The vector N is also the normal vector for both planes. Therefore the two planes are parallel.

Do you have a more convincing way to proof the two planes that contain the two individual non intersecting lines can be made parallel?

2. Mar 31, 2013

### AlephZero

Here's a different way of looking at it.

Call the two lines A and B. You can draw a line from any point on A to any point on B.

Let C be the shortest line that you can draw from a point on A to a point on B. You should be able to see that C is perpendicular to A, and also C is perpendicular to B. (For example, if the angle between A and C was not a right angle, you could draw a shorter line, from the same point on B, but making the angle closer to a right angle).

You can draw two planes perpendicular to C, one containing line A and the other containing line B, and those two planes are parallel.

3. Apr 1, 2013

### yungman

Thanks for the reply. So you can always draw two parallel planes, each contains one of the two non intersecting lines.
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