Orthogonal Vectors in a 2D Lattice with minimum area I came across an interesting problem in my research (not a mathematician). Here it goes:
Suppose, there is a 2D lattice $\Lambda$ in the X-Y plane with basis vectors $\vec{a}$ and $\vec{b}$, which are not orthogonal to each other. I want to find a rectangle in this lattice, whose area is the minimum of all possible rectangles. So far, this is what I have:
Consider two vectors in the lattice:
$\vec{v_1} = m_1 \vec{a} + n_1 \vec{b}$
$\vec{v_2} = m_2 \vec{a} + n_2 \vec{b}$
If they are orthogonal, then $\vec{v}_1 \cdot \vec{v}_2 = m_1 m_2 \left| \vec{a} \right|^2 + n_1 n_2 \left| \vec{b} \right|^2 +  (m_1 n_2 + n_2 m_1) \vec{a} \cdot \vec{b} =0 $
The area subtended by the two vectors is the norm of the cross product, i.e. $\left| \vec{v}_1 \times \vec{v}_2 \right| = \left| \vec{a}_1 \times \vec{b}_2 \right| \left|m_1n_2 - m_2n_1 \right|$
Therefore, the area is minimized if $\left|m_1n_2 - m_2n_1 \right|$ is minimized and greater than zero. And the constraint is:
$m_1 m_2 \left| \vec{a} \right|^2 + n_1 n_2 \left| \vec{b} \right|^2 +  (m_1 n_2 + n_2 m_1) \vec{a} \cdot \vec{b} =0$
I am stuck after this! May be some kind of optimization with a constraint using Lagrange multipliers? But there are too many variables and $m_1, m_2, n_1, n_2 \in \mathbb{Z}$. Any suggestions will be greatly appreciated. Thank you for your time.

THIRD answer: It turns out the condition I gave in my second answer is necessary and sufficient for existence; I can also show a two parameter family, I'm afraid to minimize could be algorithmic but not formulaic. i will have time for that aspect later. Given Gram matrix with $A,B,C$ as before,
we have the equation
$$ A \alpha \beta + B (\alpha \delta + \beta \gamma) + C \gamma \delta.  $$
So need to have $iA + jB + kC = 0,$ with $j^2 - 4ik = w^2.$
So, take integer parameters $s,t,$ then
$$ \alpha = 2i s,  \; \; \beta = 2 it,  \; \; \gamma = (j-w)s, \; \; \delta = (j+w)t.   $$ With these values, the vector $( \alpha \beta, \alpha \delta + \beta \gamma, \gamma \delta)$ is a scalar (rational) multiple of $(i,j,k),$ and we have constructed orthogonal vectors in the lattice. It is possible that $\gcd(\alpha, \beta, \gamma,\delta)> 1$ for some values of $(s,t)$ but not others. 
Furthermore, minimization of the determinant $|\alpha \delta - \beta \gamma|$ needs work, although it is a multiple of $w$ by construction. Here is a start:
$$  \alpha \delta - \beta \gamma = 4iwst.  $$ Need to think about what that means, with varying GCD's and the possibility of zero values for $s,t.$ SIGH. Added: no, if one of $s,t$ is zero, one vector in the orthogonal pair is just the zero vector, so we may rule out that possibility. Good. So, it may not be smallest, but $s=t=1$ gives an orthogonal pair. 
$$
\left(
\begin{array}{rr}
2i & j-w \\
2i & j+w
\end{array}
\right)
\left(
\begin{array}{rr}
A & B \\
B & C
\end{array}
\right)
\left(
\begin{array}{rr}
2i & 2i \\
j-w & j+w
\end{array}
\right) =
\left(
\begin{array}{cc}
4i^2 A + 4i(j-w)B + (j-w)^2C & 0 \\
0 & 4i^2 A + 4i(j+w)B + (j+w)^2C
\end{array}
\right)
$$
If $i=0,$ thus $jB + kC=0,$ we get
$$
\left(
\begin{array}{rr}
j & k \\
0 & 1
\end{array}
\right)
\left(
\begin{array}{rr}
A & B \\
B & C
\end{array}
\right)
\left(
\begin{array}{rr}
j & 0 \\
k & 1
\end{array}
\right) =
\left(
\begin{array}{cc}
j^2 A + 2jkB + k^2C & 0 \\
0 & C
\end{array}
\right)
$$
When $j=0,$ the combination of $\gcd(i,j,k)=1$ and $j^2 - 4 i k = w^2$ allows us to demand, in integers,
$$   i = x^2, \; \; \;  k = -y^2, \; \; \;  w = 2 x y, $$
with $x^2 A - y^2 C = 0.$ Then
$$
\left(
\begin{array}{rr}
x & -y \\
x & y
\end{array}
\right)
\left(
\begin{array}{rr}
A & B \\
B & C
\end{array}
\right)
\left(
\begin{array}{rr}
x & x \\
-y & y
\end{array}
\right) =
\left(
\begin{array}{cc}
A x^2 - 2 B x y + C y^2 & 0 \\
0 & A x^2 + 2 B x y + C y^2
\end{array}
\right)
$$
Anyway, these show existence for any explicit $(i,j,k)$ triple of integers such that $iA + j B + k C = 0.$ These also give upper bounds on the determinants of the change of basis matrices, again $|\alpha \delta - \beta \gamma|.$ In the case that the lattice cannot be scaled to an integral lattice, I am not confident about giving an explicit recipe for the smallest determinant that works; I suggest using these as upper bounds for a computer search. 
