When a line AB is inclined to both HP and VP, then its projection on the plane is a line shorter than its true length visible in

When a line AB is inclined to both the Horizontal Plane (HP) and the Vertical Plane (VP), the nature of its projection on these planes is defined by the principles of projection in descriptive geometry.

1. Understanding Projections:

   • The true length of a line is the actual length measured along the line itself in three-dimensional space.
   • When a line is projected onto a plane (HP or VP), the projection can often appear shorter than the true length, particularly when the line has an angle of inclination relative to the plane.

2. Inclination Effects:

   • If a line is inclined to both planes, the angles at which it meets the planes influence how long the projection will be.
   • The projection is calculated based on the cosine of the angle of inclination. Specifically, if θ is the angle of inclination, the formula for the length of the projection (L') is given by: $L' = L \cdot \cos(\theta)$ where $L$ is the true length of the line.

3. Conclusion:

   • Since $\cos(\theta)$ (where $0 < \theta < 90^\circ$) decreases as θ increases, it confirms that the projection length will indeed be shorter than the true length. This is further confirmed by vector analysis in 3D geometry.

Final Answer

The projection of line AB on both HP and VP will be shorter than its true length due to the angles of inclination affecting the projection calculations.