Exercise 11.1 answers: Introduction to Three Dimensional Geometry

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1. Any point on the x-axis is of the form (x, 0, 0).
2. The y-coordinate is the point's distance from the XZ-plane, and a point on the x-axis lies on that plane too, so y = 0.
3. The z-coordinate is the point's distance from the XY-plane, and a point on the x-axis lies on that plane too, so z = 0.

Answery-coordinate = 0 and z-coordinate = 0

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1. The y-coordinate of any point is its distance from the XZ-plane.
2. A point that lies in the XZ-plane has zero distance from that plane.
3. So its y-coordinate must be 0.

AnswerIts y-coordinate is 0.

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1. Read off the sign (+ or −) of the x, y and z coordinate of each point.
2. Match that sign pattern to the table of octants: I(+,+,+), II(−,+,+), III(−,−,+), IV(+,−,+), V(+,+,−), VI(−,+,−), VII(−,−,−), VIII(+,−,−).

3. Point	Signs (x, y, z)	Octant
   (1, 2, 3)	(+, +, +)	I
   (4, -2, 3)	(+, −, +)	IV
   (4, -2, -5)	(+, −, −)	VIII
   (4, 2, -5)	(+, +, −)	V
   (-4, 2, -5)	(−, +, −)	VI
   (-4, 2, 5)	(−, +, +)	II
   (-3, -1, 6)	(−, −, +)	III
   (-2, -4, -7)	(−, −, −)	VII

AnswerI, IV, VIII, V, VI, II, III, VII (in the order the points are given)

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(i) The x-axis and y-axis taken together determine a plane

1. A plane is named after the two axes that lie inside it.
2. The x-axis and the y-axis together lie in the XY-plane.

AnswerXY-plane

(ii) The coordinates of points in the XY-plane are of the form

1. Every point in the XY-plane has zero distance from that plane, so its z-coordinate is 0.
2. The x-coordinate and y-coordinate can be anything.

Answer(x, y, 0)

(iii) Coordinate planes divide the space into______ octants.

1. The three coordinate planes cross each other and cut all of space into pieces called octants.
2. There are 2 choices of sign (+ or −) for each of x, y and z, giving 2 × 2 × 2 pieces.

Answereight

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