Is there a collectionwise normal topological vector space which is not paracompact?

How do I assign a temporary plane or vector or reference frame in MATLAB to find the angle between this and a given vector in 3D space?

• I can't get this clearly in my head. I have a bunch of vectors which move around over time. After every instant I wish to check by what angle(s) have the vectors moved/rotated in space with respect to an initial reference/temporary plane (static). These vectors might be getting translated too over time. If it makes it easier, an analogy would be  your arm being the given vector and you moving your arm around in space while walking. You wish to find the angle by which your arm moves in space (w.r.t a reference plane/vector). So the arm can have different angles, and I wish to find those. The following figures might give a better idea about the different possible angles? How do I go about defining a temporary plane/vector to find this angle. I can't seem to understand this clearly. Any suggestions? I initially thought of it as a simple conversion from cartesian to spherical coordinates, but I need to know how to select the initial reference plane/vector.

• Answer:

  You shouldn't need a temporary plane, right? What do you want the angle to be determined against? If you just want to know "how much has the angle changed since the last time", just use the previous vector and compare it to the current one. If you just want to know "how much the vector has ever changed", compare it to the initial state. The angle itself you can just get from the dot product of the vectors, since a⋅b=|a||b|cosθa⋅b=|a||b|cos⁡θ a \cdot b = |a| |b| \cos \theta .