# Math Help - parameterized curves that lie in R3

1. ## parameterized curves that lie in R3

which image of the given parameterized curves lie on the unit sphere in R3?

1. r(x)=(sin(3x), 0, cos(3x))
2. r(x)=(x^2, cos x, sin x)
3. r(x)=(1/sqrt.2, (-cos x)/sqrt.2, (sin x)/sqrt.2)
4. r(x)= (sec x, 1/sqrt.2, cos x)

How do you actually go about determining this?

2. For a given parametrized curve $r(t)=(x(t),y(t),z(t))$ the image lie on the unit sphere of $\mathbb{R}^3$ iff $x(t)^2+y(t)^2+z(t)^2=1$ for all $t$ . For example $r(t)=(\sin 3t,0,\cos 3t)$ satisfies it because $\sin^2 3t+0^2+\cos^2 3t=1$ for all $t$.

Fernando Revilla