How to Directly Calculate the Clockwise Angle Between Two Vectors?

Direct Computation of Clockwise Angle Between Vectors

Traditionally, calculating the angle between vectors involves using the dot product, which yields the inner angle between 0° and 180°. However, this approach necessitates the use of conditional statements to determine the actual clockwise angle.

Two-Dimensional Case

In 2D, a straightforward approach exists:

dot = x1*x2 + y1*y2      # Dot product between [x1, y1] and [x2, y2]
det = x1*y2 - y1*x2      # Determinant
angle = atan2(det, dot)  # atan2(y, x) or atan2(sin, cos)

The determinant is proportional to the sine of the angle, complementing the dot product's relationship with cosine. The orientation of the angle aligns with that of the coordinate system, with positive angles indicating clockwise rotation in a left-handed system (e.g., computer graphics). Swapping the inputs changes the sign of the angle.

Three-Dimensional Case

In 3D, the angle of rotation is defined by the axis perpendicular to both vectors involved. One convention assigns positive angles to rotations that align the axis with a positive angle. Using this convention:

dot = x1*x2 + y1*y2 + z1*z2    # Between [x1, y1, z1] and [x2, y2, 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

Another special case occurs when the vectors lie within a plane with a known normal vector _n_. Adapting the 2D computation, we account for _n_:

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)

Note that n must have unit length.

Triple Product Form

The determinant can also be expressed as a triple product:

det = n · (v1 × v2)

This formula provides an alternative perspective, with the cross product being proportional to the sine of the angle and lying perpendicular to the plane, effectively aligning with _n_. The dot product then measures the length of the resulting vector with the correct sign.

Range 0° – 360°

Most atan2 implementations return angles in the range [-π, π] radians or [-180°, 180°] degrees. For positive angles [0°, 360°], add 2π to any negative results. Alternatively, atan2(-det, -dot) π can be used unconditionally for positive angles.

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