Two lines , at right angle to one another , lie on a plane which is inclined at angle to the horizontal . If the 2 lines intersect at a point on the horizontal and their angles of inclination to the horizontal are and respectively , prove that
Two lines , at right angle to one another , lie on a plane which is inclined at angle to the horizontal . If the 2 lines intersect at a point on the horizontal and their angles of inclination to the horizontal are and respectively , prove that
Hello thereddevilsDrawing a good 3-D diagram is a great help here, so do the following:
- Draw a wedge with a horizontal rectangular base PQRS, and a sloping rectangular face PABQ, so that the line AB is parallel to PS, with A vertically above Q and B vertically above R.
Then the angle that the sloping face makes with the horizontal .
- Now mark a point O on PS. Join O to A, B, Q, R. Suppose that OA and OB are the two perpendicular lines referred to in the question. Then the angles these lines make with the horizontal are:
and
Then all the triangles are right-angled (except ROQ). Suppose that . Then we have:
Similarly
But
Multiply through by , and divide by :
, assuming
, assuming
Phew!
Grandad