Unit vector tangent to a unit sphere

Question

What is an expression for a unit vector that is tangent to a unit sphere, in terms of Cartesian unit vectors?

Answer 1

Define

\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {\left(x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}

Any vector of the form

$$ \hat{\mathbf{u}} = \frac{a \hat{\boldsymbol {\theta }} +b\hat{\boldsymbol {\phi }}}{\sqrt{a^2 + b^2}} $$

is tangent to the sphere $x^2 + y^2 + z^2 = {\rm const}$