January 21st, 2017, 10:07 PM  #1 
Newbie Joined: Jan 2017 From: Pakistan Posts: 4 Thanks: 0  How to prove this?
show that in symmetric form of the line through point P'=(x',y',z') and parallel to non zero vector u = (a,b,c) is (xx')/a = (yy')/b = (zz')/c.

January 21st, 2017, 11:42 PM  #2 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16 
This is just a way to state that PP' = ku, k real, which is equivalent to saying that the line (P,P') is parallel to u. If you want to impress your teacher, you can say that this is a modern version of Thales theorem.

January 28th, 2017, 04:26 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,582 Thanks: 668 
Vector <p, q, r> is parallel to vector <u, v, w> if and only if there exist a nonzero number k such that p= ku, q= kv, and r= kw. Taking (x, y. z) to be any point on the line, with (x', y', z') the given point, then <x x', y y', z z'> is a vector in the direction of the vector (a, b, c) so that there exist a nonzero number k such that x x'= ka, y y'= kb, and z z'= kc. from that , , and . Since those are all equal to k they are equal to each other: 

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