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PAU Madrid 2025 · Angle between two lines
Statement. Lines f and h meet at P(4, 4, 2). f has direction (1, 0, 1) and h direction (1, 1, 0). Find the angle they form and justify the construction. (Adaptation of the original statement: the exam gives the figure; the coordinates are fixed here so it can be built in the editor.)
Step-by-step solution
- Draw both lines with their two projections: f'-f'' and h'-h''.
- The angle is not measured on the projections: both directions must be seen in true length.
- Draw through P a parallel to each direction if they do not already meet there; here both pass through P.
- Contain both lines in a plane (the two directions define it) and rotate it onto the horizontal plane.
- In the rotation both lines appear in true length: the angle between them is the one asked for.
Result
The angle between f and h is 60 degrees.