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Let $S$ be unit sphere in $\mathbb R^3$ center at $O(0,0,0)$. Let $A=(x_1,y_1,z_1),B = (x_2,y_2,z_2)$ be two points lying on the sphere $S$. Let $M$ be center of $AB$ which lies on the geodesics $AB$. Find the coordinate of P on sphere which lie on plane perpendicular with $AB$ and angle $(MOP) = \alpha$.

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I know how to find the coordinate of the point $M$ but have not idea to find coordinate of $P$ in term of coordinate $A,B$ and angle $\alpha$. Can anyone give me a hint? Many thanks

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The cross-product $(B-M)X(O-M) $ will produce a vector perpendicular to $MB$ and $MO$.

Adjust its length to make $\frac{|PM|}{|MO|} =\tan(\alpha) $.

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  • $\begingroup$ but it does not give the coordinate of P, and OPM is not a right angle $\endgroup$ – David Li Nov 28 '15 at 10:05

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