How to Calculate the Clockwise Angle Between Two Vectors: Beyond the Dot Product?

Calculating the Clockwise Angle Between Vectors: Beyond Classic Approaches

Classic methods for computing the angle between two vectors rely on the dot product, which only provides the inner angle (0-180 degrees). To obtain the desired clockwise angle directly, we explore alternative approaches that harness the determinant.

2D Vectors

In the 2D domain, the determinant represents a value proportional to the sine of the angle. Thus, we can calculate the angle using:

dot = x1*x2 + y1*y2 
det = x1*y2 - y1*x2 
angle = atan2(det, dot)

The resulting angle aligns with the coordinate system orientation, yielding positive values for clockwise rotations. Swapping the input vectors reverses the sign.

3D Vectors

For 3D vectors, where the axis of rotation remains undefined, we typically opt for positive angles. The normalized dot product provides an appropriate measure:

dot = x1*x2 + y1*y2 + z1*z2
lenSq1 = x1*x1 + y1*y1 + z1*z1
lenSq2 = x2*x2 + y2*y2 + z2*z2
angle = acos(dot/sqrt(lenSq1 * lenSq2))

Plane Embedded in 3D

If the vectors lie within a plane with a known normal vector n, we can leverage this information to refine the calculations:

dot = x1*x2 + y1*y2 + z1*z2
det = x1*y2*zn + x2*yn*z1 + xn*y1*z2 - z1*y2*xn - z2*yn*x1 - zn*y1*x2
angle = atan2(det, dot)

Range 0 – 360°

Common atan2 implementations return angles in the range [-π, π] radians. To obtain angles in the desired range [0, 2π] radians, simply add 2π to negative results. Alternatively, use atan2(-det, -dot) π unconditionally.

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