The German mathematician Gauss proved that the densest way to pack circles with the same radius in the plane is to place the centers of the circles on a hexagonal grid (see figure). Some molecular structures use this packing or its three-dimensional analog. Assume all circles have a radius of 1 and let r_ij be the vector that extends from the center of circle i to the center of circle j, for i, j = 0, 1,  . . . , 6.

a. Find r_0j, for j = 1, 2, . . . , 6.

b. Find r₁₂, r₃₄, and r₆₁.

c. Imagine circle 7 is added to the arrangement as shown in the
figure. Find r₀₇, r₁₇, r₄₇, and r₇₅.

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УА 3 4 6