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Feasible inner products with two unit vectors

For linearly independent unit vectors

y, z

with

c = y \cdot z

, a pair

p = x \cdot y, q = x \cdot z

is attained by some unit vector

x

exactly when

\frac{p ^{2} - 2 c pq + q ^{2}}{1 - c ^{2}} \leq 1.

The left-hand side is the squared norm of the least-norm vector having those two inner products.

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