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April 22nd, 2018, 04:23 AM  #1 
Newbie Joined: Apr 2018 From: Australia Posts: 1 Thanks: 0  Bases of subspaces in $R^3$
How do I find the bases of the set of vectors on the line x/2 = y/3 = z/4?

April 22nd, 2018, 11:19 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,165 Thanks: 867 
$\displaystyle \frac{x}{2}= \frac{y}{3}= \frac{z}{4}$ Because this is a line, it is one dimensional which means a basis consists of a single vector. Taking x= y= z= 0, that condition becomes 0= 0= 0 which is true so (0, 0, 0) is a point on that line. Taking x= 2, y= 3, z= 4, that condition becomes 1= 1= 1 which is true so (2, 3, 4) is also a point on that line. The vector <2 0, 3 0, 4 0>= <2, 3, 4> is a vector in the direction of that line so {<2, 3, 4>} is a basis for that line (more correctly, "is a basis for the vector space determined by that line.). 

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$r3$, bases, subspaces 
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