How can you directly compute the clockwise angle between two vectors in both two and three dimensions?

Direct Method for Computing Clockwise Angle Between Vectors

When dealing with two-dimensional vectors, the clockwise angle can be computed directly using the determinant and dot product. The dot product provides information about the cosine, while the determinant reveals the sine of the angle. By utilizing the atan2 function, you can obtain the clockwise angle as follows:

dot = x1*x2 + y1*y2  # Dot product of [x1, y1] and [x2, y2]
det = x1*y2 - y1*x2  # Determinant
angle = atan2(det, dot)  # Clockwise angle

Note that the orientation of the angle aligns with the coordinate system. For a right-handed system (x right, y down), clockwise angles yield positive values. Reversing the order of inputs changes the angle's sign.

In three dimensions, where vectors may lie along arbitrary planes, the axis of rotation is also arbitrary. Commonly, angles are considered positive, with the axis oriented for a positive angle. The dot product between normalized vectors can then determine the angle:

dot = x1*x2 + y1*y2 + z1*z2  # Dot product of [x1, y1, z1] and [x2, y2, z2]
lenSq1 = x1*x1 + y1*y1 + z1*z1  # Squared length of vector 1
lenSq2 = x2*x2 + y2*y2 + z2*z2  # Squared length of vector 2
angle = acos(dot/sqrt(lenSq1 * lenSq2))  # Clockwise angle

In cases where vectors reside within a known plane, a modified 2D computation can be used by incorporating the plane's normal vector into the determinant:

dot = x1*x2 + y1*y2 + z1*z2
det = n * (v1 × v2)  # Triple product
angle = atan2(det, dot)

where n is the unit-length normal vector.

Adjustments may be necessary to ensure the angle falls within the desired range (e.g., [0, 360°]). By adding 2π to negative results or using atan2(-det, -dot) π, the angle can be adjusted accordingly.

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