Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

DIRECTION COMPONENTS

441

In each of the following exercises show that three lines with
the given direction components are parallel to a plane.
2. 3,1,2; 5,-4,3; 1,6,1.
3. -1, 1,2; 2, -1, 1; 1, 1, 8.
EXERCISES ON CHAPTER XVIII
1. Show that the triangle with vertices at the points
(1, 3, - 5), (3, 4, - 7), (2, 5, - 3) is a right triangle.
2. Prove that the points (2, - 1, 5), (3, 4, - 2), (6, 2, 2),
(5, -3, 9) are the vertices of a parallelogram.
3. Show that (2, 3, 0), (4, 5, - 1), (3, 7, 1), (1, 5, 2) are
the vertices of a square.
4. Prove that the three points A, B, C with the coordinates
(5, -2, 3), (2, 0, 2), (11,-6, 5) lie on a line by showing that
the line AB has the same direction as the line AC.
5. Show that the two points (4, -2, - 6), (- 6, 3, 9) lie
on a line with the origin.
6. Show that two points (x1, Y1, Z1), (x,, Y2, z2) lie on a line
with the origin when and only when their coordinates are
proportional: x1: Y1: Zx = X2 Y2: Z2.
7. Show that the four points A, B, C, D, with the coordinates (3, 4, 2), (1, 6, 2), (3, 5, 1), (4, 5, 0), lie in a plane by
proving that the sum of the angles which BC and CD subtend
at angle A equals the angle which BD subtends at A.
8. Find the projection, on a directed line having 2, - 3, q
as its direction cosines, of the directed line-segment joining the
origin to the point (5, 2, 4). Ans. 4.
Suggestion. Use the method of ~ 2 or employ formula (2)
of Ch. XVII, ~ 4.
9. Show that the projection, on a directed line having
cos a, cos 1, cos y as its direction cosines, of the directed linesegment joining the origin to the point (x, y, z) is
x cos a + y cos / + z cos y,