# Math Help - Vector equation proof

1. ## Vector equation proof

I really have no idea how to attack this one.

If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with the center at the origin.

Thanks

2. Originally Posted by NoFace

If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with the center at the origin.
so we have $r(t) \cdot r'(t)=0,$ which gives us: $(r(t) \cdot r(t))'=2r(t) \cdot r'(t)=0.$ thus: $||r(t)||^2=r(t) \cdot r(t)=c,$ for some constant $c.$ so the curve lies on: $x^2 + y^2 + z^2 = c. \ \ \square$