Points P and Q have position vectors (2/b) and (8/13) respectively. Given that |PQ| = 10, calculate the values of b.

Step 1: Position vectors (in m)
$\vec{OP} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad \vec{OQ} = \begin{pmatrix} 3 \\ b \end{pmatrix}$

Step 2: Vector $\vec{PQ}$ (in m)
Formula: $\vec{PQ} = \vec{OQ} - \vec{OP}$
Substitute: $\vec{PQ} = \begin{pmatrix} 3 \\ b \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
Simplify: $\vec{PQ} = \begin{pmatrix} 2 \\ b-2 \end{pmatrix}$

Step 3: Magnitude $|\vec{PQ}| = 7$ m
Formula: $|\vec{PQ}| = \sqrt{ (2)^2 + (b-2)^2 }$
Given: $\sqrt{ 4 + (b-2)^2 } = 7$

Step 4: Square both sides
Formula: $\left( \sqrt{ 4 + (b-2)^2 } \right)^2 = 7^2$
Substitute: $4 + (b-2)^2 = 49$
Simplify: $(b-2)^2 = 45$

Step 5: Solve for $b$ (in m)
Formula: $b-2 = \pm \sqrt{45}$
Substitute: $\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$
Simplify: $b = 2 \pm 3\sqrt{5}$

$\mathbf{b = 2 \pm 3\sqrt{5} \, m}$
$\boxed{2 \pm 3\sqrt{5}}$