The angle between two unit vectors $ \mathbf{u} $ and $ \mathbf{v} $ is given by $ \cos heta = \mathbf{u} \cdot \mathbf{v} $. So,

February 22, 2026

$The angle between two unit vectors $ \mathbf{u} $ and $ \mathbf{v} $ is given by $ \cos heta = \mathbf{u} \cdot \mathbf{v} $. So,$

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\cos( heta_1 - heta_2) = \cos 60^\circ = rac{1}{2}.
oxed{rac{1}{2}}
Question: A museum curator is aligning two historical astrolabes represented by vectors $ egin{pmatrix} \cos lpha \ \sin lpha \end{pmatrix} $ and $ egin{pmatrix} \cos eta \ \sin eta \end{pmatrix} $ in the plane. If the cosine of the angle between them is $ rac{1}{3} $, and $ lpha + eta = 90^\circ $, find $ \sin(2lpha) $.
\cos(lpha - eta) = rac{1}{3}.
We are given $ lpha + eta = 90^\circ $, so $ eta = 90^\circ - lpha $. Substituting,
\cos(lpha - (90^\circ - lpha)) = \cos(2lpha - 90^\circ) = rac{1}{3}.

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