Plotting vectors originating from the origin in 3D

For your problem, it is probably easiest to build the graphic out of graphics primitives rather than use a pre-made convenience function such as ListPointPlot3D.

This is one way to do it:

data = {{1, 2, 3}, {3, 4, 5}, {5, 6, 7}};

Graphics3D[Arrow[{{0, 0, 0}, #}] & /@ data]

I simply used the Arrow graphics primitive. I constructed a pure function that makes an arrow starting from the origin, and mapped it over the data.

I wasn't able to find a way to make ListPointPlot3D draw lines instead of points.

As an alternative to Szabolcs' Graphics3D, here is a slightly different way using ParametricPlot3D and a replacement rule.

data = {{1, 2, 3}, {3, 4, 5}, {5, 6, 7}};
ParametricPlot3D[data*u, {u, 0, 1}] /. Line -> Arrow