Calculus: Analytic Geometry and Calculus, with Vectors

82 Vectors and geometry in three dimensions

6 Because several of the coordinates are zero, it is relatively easy to deter-

mine fl, B, C, D such that the equation Ax + By + Cz + D = 0 is satisfied by

the coordinates of the three points (3,0,0), (0,4,0), (3,4,5). Do it and thereby

find the equation of the plane containing the three points. Ans.: 20x + 15y -

12z - 60 = 0.

7 A sphere of radius 3 has its center at the origin. Observe that it is not

easy to sketch a figure showing the plane a tangent to the sphere at the point

(2,2,1) and the point T where a intersects the z axis. Find the coordinates of T.

Hint: The plane 7r is normal to the line from the center of the sphere to the point

of tangency. fins.: (0,0,9).

8 If a, b, and c are nonzero constants, show that the equation

a+b+Z=1

is the equation of the plane which intersects the coordinate axes at the points

(a,0,0), (0,b,0), and (0,0,c). Find the scalar components of a normal to the plane.

9 Find the distance from the origin to the plane of the preceding problem.

Your answer is wrong if it does not reduce to 1/1/ when a = b = c = 1.

10 A plane n1 intersects the positive x, y, and z axes 1, 2, and 3 units, respec-

tively, from the origin. A second plane ir2 intersects each positive axis one unit

farther from the origin. Would you suppose that r1 and 72 are parallel? Find

the acute angle 0 between normals to the planes. fins.: cos 0 = 1/2916/2989.

11 Find the equation of the plane which contains the point (1,2,3) and is

parallel to the plane having the equation 3x + 2y + z - 1 = 0. Check the

answer.

(^12) Find the equation of the plane it which contains the point (1,3,1) and is
perpendicular to the line L having the equations
(1) x=t, y=t, z=t+2.
Hint: If you do not know what else to do, let t = 0 to get a point P1 on L and let
t = 1 to get another point P2 on L. Then 7r must be perpendicular to P1P2.
Your answer can be worked out neatly by solving the individual equations in
(1) for t to obtain
x-0_y-0 __z-2
1 1 1 -t'
This shows that a normal to i has scalar components 1, 1, 1 and hence that the
equation of a is
(x - 1) + (y - 3) + (z - 1) = 0
orx+y+z=5.
(^13) Find the equation of the plane r containing the point P1(1,1,1) which is
perpendicular to the line L containing the points (0,1,0) and (0,0,1). Find the
coordinates of the point P2 where L intersectsa. Then find IP1P2I Observe
that the last number (which should be ) is the distance from a vertex of a unit
cube to a diagonal of a face not contain thisvertex. Look at a figure and
discover very simple reasons why 1 < IP1P2I < 2.