My Math Forum Equivalence relation

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 January 16th, 2014, 10:09 PM #1 Newbie   Joined: Jan 2014 Posts: 16 Thanks: 0 Equivalence relation Let $\mathbb{R}^3= \{\hat{v} = (x,y,z) | x,y, z \in \mathbb{R}\}.$ Let $\hat{v}, \hat{w} \in \mathbb{R}^3 - \{\hat{0}\}$. We say that $\hat{v} \sim \hat{w}$ if there is a non-zero real number $\lambda$ with $\hat{v}= \lambda \hat{w}$. Show that $\sim$ is an equivalence relation on $\mathbb{R}^3 - \{\hat{0}\}.$ What are the equivalence classes? I am not sure how to do this particular problem. I know I must show reflexive, symmetry and transitivity but would reflexive work here? Also the second part confuses me as well. How can I prove this? Thank you!
 January 17th, 2014, 03:35 PM #2 Global Moderator   Joined: May 2007 Posts: 6,766 Thanks: 698 Re: Equivalence relation Reflexive: v ~ ?v, when ? = 1. Symmetric: v ~ ?w then w ~ (1/?)v, since ? ? 0.
 January 17th, 2014, 04:27 PM #3 Member   Joined: Sep 2013 Posts: 32 Thanks: 0 Re: Equivalence relation Thanks mathman!
 January 17th, 2014, 04:30 PM #4 Newbie   Joined: Jan 2014 Posts: 16 Thanks: 0 Re: Equivalence relation Thanks! I think I know what to do now with the equivalence relation. But how do I do the second part dealing with equivalence classes?

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