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EXAMPLES OF ANALYTICAL GEOMETRY OF

THREE DIMENSIONS.

I. The straight line and plane. Let OA, OB, OC be three edges of a rectangular parallelepiped which meet at a point 0; take 0 for the origin and the directions of OA, OB, OC for the axes of x, y, z respectively; complete the parallelepiped; let D be the vertex opposite to A, E that opposite to B, F that opposite to C, G that opposite to 0. Let OA=a, OB=b, 00=c; and use these data in the following Examples from 1 to 11.

1. Find the equation to the plane passing through D, E, F.

2. Find the equation to the plane passing through G, A, B.

3. Find the equation to the plane passing through G, 0, A; also the equation to the plane passing through G, O, B.

4. Find the equations to the line OG.
5. Find the equations to the lines EB and AD.

6. Find the length of the perpendicular from the origin on the plane in Example 1.

7. Find the length of the perpendicular from C on the plane in Example 2.

8. Find the angle between the planes in Example 3.

9. Find the angle between the line in Example 4, and the normal to the plane in Example 1. 10. Find the angle between the lines in Example 5.

1

T. A. G.

11. Find the equations to the line which passes through O, and the centre of the face AEGF.

12. Interpret the equation

xc+ y2 + z = (a cos a + y cos ß + z cos ry)?,

where cos'a + cos' ß + cos'y=1.

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14. Find the equation to the plane which contains the line whose equations are Ax+By+ Cz=D, A'x + B'y + C'z=D',

and the point (a, b, y). 15. Find the equation to the plane which passes through the origin and through the line of intersection of the planes

Ax+By+ Cz=D, and A'x + B'y + C'z=D'; and determine the condition that it may bisect the angle between them,

16. Find the equation to the plane which passes through the two parallel lines

x - a' _y-b -c'

-; i

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n

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17. The equation to one plane through the origin bisecting the angle between the lines through the origin, the direction cosines of which are l, m,, ng, and l, m, n,, and perpendicular to the plane containing them is

(1-7) x + (m, -m) y + (n - n,) z = 0; and the equation to the other plane is

(i+13) x + (m, + m.) y + (n + n,) z=0.

18. Shew that the equation to a plane which passes through the point (a, b, y) and cuts off portions a, b from the axes of x and y respectively is

z laß +3

C

1

+ gla

8-1}.

a

19. Find the equation to the plane which contains a given line and is perpendicular to a given plane.

20. Shew that if the straight lines

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21. Find the length of the perpendicular from the point (1, -1, 2) upon the line x=y=2z.

22. Find the equation to the plane which passes through the lines

y 5

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23. Find the equations to a straight line which passes through the point (a, b, c) and makes a given angle with the plane

Ax+By+ Cz = 0. 24. Find the equation to the plane perpendicular to a given plane, such that their line of intersection shall lie in one of the co-ordinate planes.

25. If the three adjacent edges of a cube be taken for the co-ordinate axes, find the co-ordinates of the points in which a plane perpendicular to the diagonal through the origin and bisecting that diagonal will meet the edges.

26. Determine the plane which contains a given straight line, and makes a given angle with a given plane.

27. A straight line makes an angle of 60° with one axis, and an angle of 45° with another; what angle does it make with the third axis ?

28. Interpret x2 = = zo.

29. Find the condition that must hold in order that the equations

x=cy + bz, y=az + cx,

: bx + ay

may represent a straight line; and shew that the equations to the line then are

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30. Through the origin and the line of intersection of the planes

x cos a + y cos ß + z cos y p=0,

and w cos Q, + y cos ß. +2 cos Y - P2=0, a plane is drawn; perpendicular to this plane and through its line of intersection with the plane of (@c, y) another plane is drawn; find its equation.

31. Find the equation to the plane which passes through a given point and is perpendicular to the line of intersection of two given planes.

32. From any point P are drawn PM, PN perpendicular to the planes of (z, x) and (z, y); if O be the origin, a, b, y, ө the angles which OP makes with the co-ordinate planes and the plane OMN, then will

cosec 0 = coseca + cosecB + cosecʻy.

33. Apply the equation to a plane

x cos a + y cosB +z cos y=8 to prove

the following theorem; a triangle is projected on each of three rectangular planes; shew that the sum of the pyramids which have these projections for bases and a common vertex in the plane of the triangle is equal to the pyramid which has the triangle for base and the origin for vertex.

34. Find the equations to the straight line joining the points (a, b, c) and (a', b', c'); and shew that it will

pass through the origin if aa'+bb' + cc' = pp', where p and p' are the distances of the points respectively from the origin. 35. Express the equations to the line

a,x+bay+= 2,2 +b2y +z = azxc + b3y +cz in the form

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represent four straight lines, and that the angle between any two of them = cos?

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37. The equations to two planes are

lx + my + nz = p, 1'& + m'y + n'z = p', where

1? + m2 + n =1, 1" + m" +n" = 1; find the lengths of the perpendiculars from the origin upon the two planes which pass through their line of intersection and bisect the angles between them.

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