[Steps Shown] One common parametrization of the sphere of radius 1 centered at the origin is r(u, v)= sin u cos v \vecimath+ sin u sin v \vecjmath+ cos u

Question: One common parametrization of the sphere of radius 1 centered at the origin is

\[\vec{r}(u, v)=\sin u \cos v \vec{\imath}+\sin u \sin v \vec{\jmath}+\cos u \vec{k}\]

Find formulas for the two normals to the sphere at parameter values $(u, v)$. The algebra will look a little bit intimidating, but things actually simplify nicely, particularly if $\sin u$ is factored from each component of the normal vectors, and the remaining vector portion is compared to the original $\vec{r}(u, v)$.

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[Steps Shown] One common parametrization of the sphere of radius 1 centered at the origin is r(u, v)= sin u cos v \vecimath+ sin u sin v \vecjmath+ cos u

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