When a parallelogram of two vectors is a rectangle

It is given that \(\mathbf a=\begin{pmatrix}3\\ 4\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}k\\ 2\end{pmatrix}\), where \(k\) is a real constant. The vectors \(\mathbf a\) and \(\mathbf b\) are drawn from the same point \(O\) as adjacent sides of a parallelogram \(OACB\).

1. Show that \(|\mathbf a+\mathbf b|\ge6\) for every value of \(k\), state the value of \(k\) for which equality holds, and give a geometric reason why \(|\mathbf a+\mathbf b|\) can never be less than \(6\).[3]
2. Find the value of \(k\) for which the diagonals \(OC\) and \(AB\) of the parallelogram have equal length.[3]
3. For this value of \(k\), show that \(OA\) is perpendicular to \(OB\) by comparing the gradients of \(\mathbf a\) and \(\mathbf b\), and explain what kind of parallelogram \(OACB\) is.[2]