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I am having really two simple questions, but following two things are confusing me.

Question 1

If I know plane parameter (v3) of a given plane (say AB); if a pair of rays are incident at a point on the plane (say, point - O), how can I compute their incident angles (i.e. POC and POD), using v1, v2, v3? Where v1, v2 are vectors along the incident rays.

enter image description here

Question 2

Similarly, if I have many pairs of incident rays falling onto this plane, then, how can I really identify which pairs are making large angles and which are not (i.e. DOC angle). I noticed that to get DOC angle some time, I have to add both incident angles and some time have to subtract them.

enter image description here

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up vote 1 down vote accepted

The acute angle between two vectors can be defined using their inner product: $$ \cos\angle POC = \frac{|\mathbf{v}_1\cdot \mathbf{v}_3|}{\|\mathbf{v}_1\|\|\mathbf{v}_3\|}, $$ notice here an absolute value is added because $\mathbf{v}_1\cdot \mathbf{v}_3$ may be negative (actually negative based on your picture).

Similarly, $$ \cos\angle POD = \frac{|\mathbf{v}_2\cdot \mathbf{v}_3|}{\|\mathbf{v}_2\|\|\mathbf{v}_3\|}. $$

If you want to find the maximum of the acute angles between many pairs of incident rays, just find whose inner product is closest to $0$ (near perpendicular).

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