Step 1: Use the squared-length condition:
Unit length means $U_x^2 + U_y^2 + U_z^2 = 1$.
Step 2: Substitute:
$U_x^2 = 0.81$ and $U_y^2 = 0.04$, so $U_x^2 + U_y^2 = 0.85$.
The remaining share is $m^2 = 1 - 0.85 = 0.15$.
Step 3: Interpret:
Only $\sqrt{0.15}$ in option (B) has this square. The number 0.85 in option (A) is the sum of the other two squares, which is a tempting trap. It is not the value of $m$ itself.
Final Answer:
$m = \sqrt{0.15}$, option (B).
\[ \boxed{\sqrt{0.15} \text{ (B)}} \]