= Solution
A <spherical tensor operator> of rank $k$ transforms under <rotation> like an <angular momentum> $k$. The <Wigner-Eckart theorem> writes its <matrix elements> as
$$
\langle\lambda'j'm'|T_q^{(k)}|\lambda jm\rangle
=\frac{\langle jm;kq|j'm'\rangle}{\sqrt{2j'+1}}\,
\langle\lambda'j'\Vert T^{(k)}\Vert\lambda j\rangle.
$$
The <reduced matrix element> contains the radial and other dynamical information; the <Clebsch-Gordan coefficients> determine the dependence on the projections. Ratios within a fixed pair of multiplets therefore follow without a radial integral. The angular <selection rules> include $m'=m+q$ and $|j-k|\leq j'\leq j+k$.
For the position vector, choose the rank-one components
$$
T_0=z,\qquad T_{\pm1}=\mp\frac{x\pm iy}{\sqrt2},\qquad
x=\frac{T_{-1}-T_{+1}}{\sqrt2}.
$$
Let $R=\langle\lambda',3/2\Vert T^{(1)}\Vert\lambda,1/2\rangle$. The stretched <Clebsch-Gordan coefficients> and one lowering step give
$$
\left\langle\frac12,\frac12;1,1\middle|\frac32,\frac32\right\rangle=1,\qquad
\left\langle\frac12,\frac12;1,0\middle|\frac32,\frac12\right\rangle=\sqrt{\frac23}.
$$
For example, apply the total <angular momentum ladder operator> to the stretched state. The result has components with coefficients $1/\sqrt3$ and $\sqrt{2/3}$, corresponding to lowering the initial <angular momentum> and the vector component, respectively.
Only the $q=+1$ component of $x$ contributes to $M_1$. The $q=0$ operator $z$ cannot connect the projections in $M_2$, so that <matrix element> vanishes. With the convention above,
$$
M_1=-\frac{R}{2\sqrt2},\qquad M_2=0,\qquad M_3=\frac{R}{\sqrt6}.
$$
Thus the requested relative magnitudes and their squared values are
$$
\boxed{|M_1|:|M_2|:|M_3|=\sqrt3:0:2},\qquad
\boxed{|M_1|^2:|M_2|^2:|M_3|^2=3:0:4}.
$$
These ratios presume a common nonzero <reduced matrix element>. The position operator has odd <parity>, so an <electric dipole transition> requires opposite-parity multiplets; for one <Electron> the <electric-dipole selection rule> also gives $\Delta\ell=\pm1$. The displayed $j,m$ labels alone do not establish that condition. If the common <reduced matrix element> vanishes, all three <matrix elements> vanish and no nonzero ratio can be inferred.
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