Reflect the origin across an equal-radius chord

The points $A$ and $B$ have position vectors $\mathbf a$ and $\mathbf b$ respectively, relative to the origin $O$ such that $|\mathbf a|=|\mathbf b|$. The point $P$ with position vector $\mathbf p$ lies on $AB$ such that $\mathbf b\cdot\mathbf p=\mathbf a\cdot\mathbf p$.

(i) Show that $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{OP}$.

(ii) Determine the position vector of the point $D$ in terms of $\mathbf a$ and $\mathbf b$, where $D$ is the reflection of $O$ about the line $AB$.

(iii) Give the geometrical meaning of $|\mathbf a\times\mathbf b|$.