that's why i said that the normal line to the surface should be used when you calculate the angle between the surface with a random line. any line that's drawn on the surface is totally useless for the calculation and it only leads to confusion and sign.

You're correct in noting that when calculating the angle between a surface and a line (or vector), the most effective approach is to use the normal vector to the surface. The normal vector is perpendicular to the tangent plane of the surface at that point, and it provides a clear reference for measuring angles. Here’s a quick outline of the concept: 1. **Normal Vector**: Given a surface, the normal vector at a point provides a direction that is orthogonal (perpendicular) to the surface at that point. 2. **Angle Calculation**: To find the angle \( \theta \) between a line (or vector) \( \mathbf{v} \) and the surface, you can use the dot product formula: \[ \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{n}}{|\mathbf{v}| |\mathbf{n}|} \] where \( \mathbf{n} \) is the normal vector to the surface. The angle \( \theta \) is then defined such that \( \theta \) is the angle between the line (vector) and the normal vector. 3. **Angle with Surface**: To find the angle between the line and the surface itself, you would calculate \( 90^\circ - \theta \), since the normal vector is orthogonal to the surface. Focusing on the normal vector is indeed beneficial because it avoids complications that could arise from assessing angles using tangent lines on the surface itself. Any deviation from using the normal vector could introduce ambiguity or lead to miscalculation, particularly in more complex geometries.