Problem $2\mathrm{.}5.$ Assume vec$(c)(t)$ is a smooth parametrized curve.

$($a$)[2$ pts$]$ If vec$(c)(t)$ lies on the surface of the sphere of radius $R>0$ for all tinR, show that the velocity vec$(c{)}^{\text{'}}(t)$ is always orthogonal to the position vec$(c)(t).$

Hint: the sphere of radius $R>0$ can be described as all vectors with $||vec(x)|{|}^2=R^2.$ Plug in vec$(c)(t)$ and do implicit differentiation in $t.$

$($b$)[1$ pt$]$ Conversely, show that if the position and velocity are always orthogonal, vec$(c)(t)*vec(c{)}^{\text{'}}(t)=0$ for all tinR, then the path vec$(c)(t)$ always remains on a sphere.