(a) To solve the equations \(u + 2v = 2i\) and \(iu + v = 3\), we can express \(u\) and \(v\) in terms of real and imaginary parts. Let \(u = x + yi\) and \(v = a + bi\).

Substitute into the first equation:

\((x + yi) + 2(a + bi) = 2i\)

Equating real and imaginary parts gives:

\(x + 2a = 0\)

\(y + 2b = 2\)

Substitute into the second equation:

\(i(x + yi) + (a + bi) = 3\)

\((-y + xi) + (a + bi) = 3\)

Equating real and imaginary parts gives:

\(a - y = 3\)

\(x + b = 0\)

Solving these equations, we find:

\(x = -2, y = -2, a = 1, b = 2\)

Thus, \(u = -2 - 2i\) and \(v = 1 + 2i\).

(b) The locus \(|z + i| = 1\) is a circle centered at \(-i\) with radius 1. The locus \(\text{arg}(w - 2) = \frac{3}{4}\pi\) is a half-line starting from 2 at an angle of \(\frac{3}{4}\pi\) to the real axis.

The least value of \(|z - w|\) is found by considering the distance from the center of the circle to the line, which is \(\frac{3}{\sqrt{2} - 1}\) or approximately 1.12.