Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

THE POINT, LINE, AND PLANE

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both to the given plane and to H, it must be perpendicular to
their intersection, which is the horizontal trace of the given plane.
Since the horizontal trace of the given plane is perpendicular to
the horizontal projecting plane of the given line, it must be perpendicular to any line of the horizontal projecting plane passing
through its foot, as the horizontal projection of the given line.
By a similar course of reasoning in connection with the vertical projecting plane of the given line we may prove that the
vertical projection of the given line must be perpendicular to
the vertical trace of the given plane.
45. Proposition 2. If the two projections of a straight line are perpendicular respectively to the two traces of a given.plane, the line
itself will, in general, be perpendicular to the plane.
Proof. The horizontal projecting plane of the line is perpendicular to the horizontal trace of the given plane and therefore
perpendicular to the plane. The vertical projecting plane of the
line is perpendicular to the vertical trace of the given plane and
therefore perpendicular to the plane. Since the horizontal and
vertical projecting planes of the line are both perpendicular to the
given plane, their intersection, which is the line itself, must be
perpendicular to the plane.
46. Problem 13. Draw the traces of a plane which is perpendicular
to H and makes an angle of 45 degrees with V.
47. Problem 14. Draw the traces of a plane which is perpendicular
to V and makes an angle of 60 degrees with H.
48. Problem 15. Given a plane which makes an angle of 30 degrees
with H, and whose horizontal trace is parallel to G-L and 4 units
back of V; required to draw the vertical trace.
49. Problem 16. Draw the traces of a plane parallel to the plane
determined in Problem 15.
50. Problem 17. Given a plane which is oblique to G-L, whose
horizontal trace is back of V, and whose vertical trace is above H;
required to draw the projections of a straight line in this plane.
51. Problem 18. Given a plane which is parallel to G-L; required
to draw the projections of a point in this plane.