= Solution
Use the <Fourier transform> convention $\widehat\phi(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}\phi(x)dx$. For every <orthogonal matrix> $Q$, changing variables $x=Qy$ gives
$$
\widehat\phi(Q\xi)=\int e^{-iy\cdot\xi}\phi(Qy)dy=\widehat\phi(\xi),
$$
since the Jacobian has absolute determinant one and a radial function satisfies $\phi(Qy)=\phi(y)$. Any two vectors of the same length are related by an <orthogonal transformation>, so $\boxed{\widehat\phi\text{ is radial}}$.
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