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A certain crystal, $X$, is formed from two types of atom, $A$ and $B$. The atoms of $A$ are found at all the points, and only the points, with coordinates $(l, m, n)$ for any whole numbers $l, m, n$; the atoms of $B$ are found at all the points, and only the points, with coordinates $(l+0.5, m+0.5, n+0.5)$.
Think about the geometry of this crystal. Can you visualise its structure? Can you devise a clear pictorial representation? How simply can you describe its structure in words?
How close are the various $A$ and $B$ atoms to each other? What bond angles are formed?
What crystal structure does this represent?
Can you represent any other crystal structures in a similar way?
A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?
As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.