Let T={(x,y,z) in R^3 | (x,y,z) ≠ (0,0,0)}. Define ~ on T by (x1, y1, z1) ~ (x2, y2, z2) if there exists a nonzero real number β such that x1 = βx2, y1 = βy2 and z1 = βz2. Show that ~ is an equivalence relation on T.
I know to prove equivalence relation you need to prove that reflexive, symmetric and transative all exists. I am not sure how to prove them. Any help/suggestions would be great.