Show that the vector field $X(x, y, z)=(xy-z^2, yz-x^2, x^2+z^2+xz-1)$ is tangent to the set $x^2 + y^2 + z^2 = 1$ I know I need to find functions $F(t)$, $G(t)$, and $H(t)$ such that $F(0)=x$, $G(0)=y$, and $H(0)=z$ and $F'(0)=xy-z^2$, $G'(0)=yz-x^2$, and $H'(0)=x^2+z^2+xz-1$.  It's also necessary that $(F(t))^2 +(G(t))^2 + (H(t))^2 = 1$ for all $t$. Just not sure where to go from here...

We will show that the vector field is perpendicular to any point $(x,y,z)$ on sphere by considering the dot product of the vector field and the normal to the sphere at that point.
$(xy-z^2,yz-x^2,x^2+z^2+xz-1)\cdot(x,y,z)=x^2y-xz^2+zy^2-yx^2+zx^2+z^3+xz^2-z=zy^2+zx^2+z^3-z=z(x^2+y^2+z^2-1).$ We know on the surface of the sphere $x^2+y^2+z^2-1=0$. Therefore the result holds.
