Can't figure out how to prove this.

Y

Yusuke

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Mar 27, 2006
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Well, I'm working on a bonus theorem that's been given to us for the third quarter, and to prove it using the idea I have now I need to prove this first.

"Through a point outside a given plane, there is at least one plane parallel to the given plane."

Any help that can be given would be appreciated.

(Note, the bigger theorem. is "If line l is parallel to line m, line m is parallel to line n, and l, m, and n are not all coplanar, then l is parallel to n.")
 
Iff two lines are parallel their direction vectors are linear multiples.
The direction vector of l is (a<sub>0</sub>,b<sub>0</sub>,c<sub>0</sub>)
The direction vector of m is (a<sub>1</sub>,b<sub>1</sub>,c<sub>1</sub>)
The direction vector of n is (a<sub>2</sub>,b<sub>2</sub>,c<sub>2</sub>)
m is parallel to l so (a<sub>1</sub>,b<sub>1</sub>,c<sub>1</sub>) = p(a<sub>0</sub>,b<sub>0</sub>,c<sub>0</sub>)
n is parallel to m so (a<sub>2</sub>,b<sub>2</sub>,c<sub>2</sub>) = q(a<sub>1</sub>,b<sub>1</sub>,c<sub>1</sub>)
(a<sub>2</sub>,b<sub>2</sub>,c<sub>2</sub>) = pq(a<sub>0</sub>,b<sub>0</sub>,c<sub>0</sub>)
The direction vectors of l and n are linear multiples
l and n are parallel.
 
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