To find the reciprocal of a 3D vector first write a=mi+nj+pk where m, n and p are constants (not related to the position vectors in the question) and i, j and k are orthogonal unit vectors. If r is the reciprocal vector then r=xi+yj+zk.
So a•r=1 (scalar) = mx+ny+pz.
|a|^2=a^2=m^2+n^2+p^2.
And axb = 0 (vector), and nz-py=mz-px=my-nx=0.
So z=px/m and y=nx/m; mx+n^2x/m+p^2x/m=1; multiply through by m:
(m^2+n^2+p^2)x=m; x=m/a^2 and y=n/a^2 and z=p/a^2.
So r=mi/a^2+nj/a^2+pk/a^2, where m=a•i, n=a•j, p=a•k.
The reciprocal will help us to calculate the expression for the position vector for q.
Let's represent a=(a1, a2, a3) then 1/2a=(1/2(a1^2+a2^2+a3^2)(a1, a2, a3)=a/2a^2.
Similarly, 3/4b=3b/4b^2.
If the three position vectors, p, q, r are collinear, then (p+q)xr=0.
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