Location from range measurements. the 3-vector x represents a location in 3-d. we measure the distances (also called the range) of x to four points at known locations a1; a2; a3; a4:

p1 = ll x-a1 ii , p2 = ll x-a2 ii, p3 = ll x-a3 ii, p4 = ll x-a4 ii

express these distance conditions as a set of three linear equations in the vector x. hint. square the distance equations, and subtract one from the others.