Geometry Level 3

Let unit vectors \(\vec a\) and \(\vec b\) be perpedicular to each other, and a unit vector \(\vec c\) be inclined at an angle \(\theta\) to both \(\vec a\) and \(\vec b\). If \(\vec c = \alpha \vec a + \beta \vec b + \gamma (\vec a \times \vec b) \).

Precisely how many of the following equations are true?

(A): \(\alpha =\beta\).
(B): \( 1 - 2\alpha^2 = \gamma^2\).
(C): \(\alpha^2 = \frac{1+\cos2\theta}2 \).
(D): \(\alpha^2 - \beta^2= \gamma^2\).


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