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All angles are in radians.
(1) Without loss of generality take coordinate axes so that
$A$ is the point$(0,0,1)$, the xz-plane contains the point $C$ and
the yz-plane contains the point $B$.
(2) Thinking of $A$ as the North Pole then $C$ has latitude
$u$ and longitude 0 and $B$ has latitude $v$ and longitude
$\pi/2$.
(3) Find the 3D coordinates of $B$ and $C$. Where the origin O
is the centre of the sphere ${\bf OA, OB}$ and ${\bf OC}$ are
vectors of unit length.
(4) Use scalar products and vectors ${\bf OA, OB}$ and ${\bf
OC}$ to find the lengths of the arcs $AB, BC$ and $CA$ in terms of
$u$ and $v$. The required result follows.
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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.