Show that the tangent lines to the regular parametrized curve $ \alpha \left( t \right) = \left( {3t,3t^2 ,2t^3 } \right) $ make a constant angle with the line $y=0$ , $z=x$
First of all, the derivate of that curve is $ \left( {3,6t,6t^2 } \right) $ So in an arbitrary point of the curve at $ t=$ \varphi $ _0 $ the tangent line is $ \eqalign{ & \left( {3\varphi _0 ,3\varphi _0 ^2 ,2\varphi _0 ^3 } \right) + t\left( {3,6\varphi _0 ,6\varphi _0 ^2 } \right) \cr & = \left( {3\varphi _0 + 3\,t\,\,,\,\,\,3\varphi _0 ^2 + 6\,t\,\varphi _0 ,\,\,\,2\varphi _0 ^3 + 2\,t\,\varphi _0 ^2 } \right) \cr} $ The other line is $ (u,0,u) $ but the dot product betweem this two lines is not constant, what is bad?