Which of the following transformations will rotate a point $(x, y, z)$ 90 degrees counterclockwise about the Z-axis?

Correct Option: A. Solution: A 90-degree counterclockwise rotation about the Z-axis transforms a point $(x, y, z)$ to $(y, -x, z)$. This is because the rotation affects only the $x$ and $y$ coordinates, swapping them and changing the sign of the new $x$ coordinate.

If a point in space has coordinates $(x, y, z)$ where $x > 0$, $y 0$, in which octant does it lie?

Correct Option: D. Solution: In the fifth octant, the coordinates are $x > 0$, $y 0$.

Which coordinate plane is defined by the X and Y axes?

Correct Option: A. Solution: The XY-plane is defined by the X and Y axes.

Consider a point $P(x, y, z)$ located inside a rectangular prism. If the coordinates of point $P$ are $(3, -2, 5)$, which octant does this point lie in according to the given coordinate system?

Correct Option: D. Solution: In the given coordinate system, octant IV is defined by $x > 0$, $y 0$. Since point $P$ has coordinates $(3, -2, 5)$, it satisfies these conditions and thus lies in the fourth octant.

A rectangular prism has its vertices at $O(0,0,0)$, $A(4,0,0)$, $B(0,5,0)$, and $C(0,0,6)$. What is the volume of this prism?

Correct Option: A. Solution: The volume of a rectangular prism is given by the product of its length, width, and height. Here, the length is 4 units (from $O$ to $A$), the width is 5 units (from $O$ to $B$), and the height is 6 units (from $O$ to $C$). Thus, the volume is $4 \times 5 \times 6 = 120$ cubic units.

A plane in 3D space is given by the equation $2x - 3y + 4z = 12$. What is the normal vector to this plane?

Correct Option: A. Solution: The normal vector to a plane given by the equation $ax + by + cz = d$ is $(a, b, c)$. Therefore, the normal vector to the plane $2x - 3y + 4z = 12$ is $(2, -3, 4)$.

Introduction to Three Dimensional Geometry

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Summary

Chapter 11: Introduction to Three Dimensional Geometry

Summary

Three-dimensional geometry involves locating points in space using three coordinates.
Coordinates are represented as (x, y, z) corresponding to distances from three mutually perpendicular planes.
The three coordinate planes are the XY-plane, YZ-plane, and ZX-plane.
The origin in three-dimensional space is denoted as O(0, 0, 0).
The space is divided into eight octants based on the signs of the coordinates.
The distance between two points in three-dimensional space can be calculated using the formula derived from the Pythagorean theorem.
Historical contributions to three-dimensional geometry include work by Descartes, Fermat, and Euler.

Learning Objectives

Understand the concept of three-dimensional geometry.
Identify and describe the coordinate axes and coordinate planes in three-dimensional space.
Determine the coordinates of a point in space using ordered triplets (x, y, z).
Recognize the significance of the origin and octants in three-dimensional geometry.
Calculate the distance between two points in three-dimensional space.
Apply the concepts of three-dimensional geometry to solve problems involving points, lines, and shapes.

Detailed Notes

Introduction to Three Dimensional Geometry

Mathematics is both the queen and the hand-maiden of all sciences - E.T. BELL

11.1 Introduction

To locate a point in a plane, two intersecting mutually perpendicular lines (coordinate axes) are needed.
In three-dimensional space, three numbers (coordinates) are required to represent a point's position.
- Example: Position of a ball thrown in space or an aeroplane's flight path.
- Coordinates are the perpendicular distances from three mutually perpendicular planes (e.g., floor and two walls).

11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space

Three mutually perpendicular planes intersect at a point O, forming:
- X-axis (horizontal)
- Y-axis (horizontal)
- Z-axis (vertical)
These planes create the XY-plane, YZ-plane, and ZX-plane, dividing space into eight octants.

11.3 Coordinates of a Point in Space

A point P in space corresponds to an ordered triplet (x, y, z).
To locate point P:
1. Drop a perpendicular PM to the XY-plane (M is the foot of the perpendicular).
2. Draw a perpendicular ML to the x-axis (L).
3. The coordinates are defined as:
  - OA = x
  - LM = y
  - MP = z
The coordinates of the origin O are (0, 0, 0).

11.4 Distance between Two Points

For points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):
- The distance formula is derived from the Pythagorean theorem in three dimensions:
  - PQ² = PA² + AQ²
  - Where PA and AQ are the lengths along the respective axes.

Example Problems

Example 1: If P is (2, 4, 5), find the coordinates of F.
- Solution: F's coordinates are (2, 0, 5).
Example 2: Determine the octant for points (-3, 1, 2) and (-3, 1, -2).
- Solution: (-3, 1, 2) is in the second octant; (-3, 1, -2) is in octant VI.

Table of Octants

Coordinates	I	II	III	IV	V	VI	VII	VIII
x	+	-	-	+	+	-	-	+
y	+	+	-	-	+	+	-	-
z	+	+	+	+	-	-	-	-

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Three Dimensional Geometry

Common Pitfalls

Misunderstanding Coordinates: Students often confuse the coordinates of points in three-dimensional space. Remember that a point is represented as (x, y, z), where each value corresponds to a distance from the respective coordinate planes.
Octant Confusion: Identifying the correct octant for a point can be tricky. Ensure you understand the signs of x, y, and z coordinates to determine the octant accurately.
Distance Formula Errors: When calculating the distance between two points, students may forget to apply the three-dimensional distance formula correctly. The formula is given by PQ² = PA² + AN² + NQ².

Tips for Success

Visualize the Geometry: Use diagrams to visualize points, lines, and planes in three-dimensional space. This can help in understanding the relationships between different elements.
Practice with Examples: Work through examples, such as finding the coordinates of points or determining the lengths of segments in three-dimensional figures.
Review Coordinate Systems: Familiarize yourself with the three coordinate planes (XY, YZ, ZX) and how they divide space into octants. Understanding this will aid in solving problems related to coordinates.
Check Your Work: Always double-check your calculations, especially when determining distances or coordinates, to avoid simple arithmetic mistakes.

Practice & Assessment

Multiple Choice Questions

abc

ab + bc + ca

a + b + c

a^2 + b^2 + c^2

Correct Answer: A

Solution:

The volume of a rectangular prism with side lengths

a

b

, and

c

is given by the product

abc

. Thus, the volume is

abc

(4, -2, 2)

(2, -6, 8)

(3, -4, 5)

(4, -2, 8)

Correct Answer: A

Solution:

The translation of a point

(x, y, z)

by a vector

(a, b, c)

results in new coordinates

(x+a, y+b, z+c)

. Therefore, the new coordinates of

Q

are

(3+1, -4+2, 5-3) = (4, -2, 2)

(4, -2, -3)

(4, 2, 3)

(4, -2, 3)

(4, 2, -3)

Correct Answer: A

Solution:

Reflecting a point across the XY-plane changes the sign of the

z

-coordinate. Thus, the coordinates of

P

after reflection are

(4, -2, -3)

(x, y, z) \rightarrow (-x, y, z)

(x, y, z) \rightarrow (x, -y, z)

(x, y, z) \rightarrow (x, y, -z)

(x, y, z) \rightarrow (-x, -y, -z)

Correct Answer: A

Solution:

To reflect a point across the YZ-plane, the

x

-coordinate changes sign while the

y

and

z

coordinates remain the same. Thus, the transformation is

(x, y, z) \rightarrow (-x, y, z)

Point in the first octant

Point in the fourth octant

Point in the fifth octant

Point in the eighth octant

Correct Answer: A

Solution:

In the first octant, all coordinates (

x

y

z

) are positive.

First Octant

Second Octant

Fourth Octant

Fifth Octant

Correct Answer: C

Solution:

In a 3D coordinate system, the octants are determined by the signs of the coordinates

(x, y, z)

. For the point

(2, -3, 4)

x

is positive,

y

is negative, and

z

is positive, which corresponds to the Fourth Octant.

60 cubic units

12 cubic units

20 cubic units

15 cubic units

Correct Answer: A

Solution:

The volume of a rectangular prism is given by the product of its length, width, and height. Here, the dimensions are

4

3

, and

5

. Thus, the volume is

4 \times 3 \times 5 = 60

cubic units.

(y, -x, z)

(-y, x, z)

(-x, -y, z)

(x, y, z)

Correct Answer: A

Solution:

A 90-degree counterclockwise rotation about the Z-axis transforms a point

(x, y, z)

(y, -x, z)

. This is because the rotation affects only the

x

and

y

coordinates, swapping them and changing the sign of the new

x

coordinate.

(0, 0, 0)

(0, 5, 0)

(5, 0, 0)

(0, 0, 5)

Correct Answer: B

Solution:

The point (0, 5, 0) lies on the Y-axis because its x and z coordinates are zero.

Point in the second octant

Point in the third octant

Point in the sixth octant

Point in the seventh octant

Correct Answer: A

Solution:

In the second octant, the

x

coordinate is negative, and both

y

and

z

coordinates are positive.

Correct Answer: D

Solution:

In the fifth octant, the coordinates are

x > 0

y < 0

, and

z > 0

72 cubic units

24 cubic units

12 cubic units

48 cubic units

Correct Answer: A

Solution:

The volume of a rectangular prism is given by the product of its length, width, and height. Here, the dimensions are 3, 4, and 6, so the volume is

3 \times 4 \times 6 = 72

cubic units.

-2

None of these

Correct Answer: C

Solution:

In 3D geometry, the Z-coordinate represents the height from the floor.

Correct Answer: C

Solution:

The top face of the prism includes the point C, as described in the diagram.

(1, 2, 3)

(-1, 2, 3)

(1, -2, 3)

(1, 2, -3)

Correct Answer: A

Solution:

The point (1, 2, 3) has all positive coordinates as all values for x, y, and z are positive.

To locate the point on a plane

To determine the point's distance from the origin

To represent the point's position in three-dimensional space

To calculate the point's velocity

Correct Answer: C

Solution:

Three numbers are used to represent the perpendicular distances of the point from three mutually perpendicular planes, thus locating its position in three-dimensional space.

On the X-axis

On the Y-axis

On the Z-axis

At the origin

Correct Answer: D

Solution:

The point (0, 0, 0) is at the origin, where all three coordinate planes intersect.

Angle at point P

Angle at point Q

Angle at point N

Angle at point A

Correct Answer: C

Solution:

The angles at points N and A are marked as 90° in the diagram.

Point P

Point Q

Point O

Point N

Correct Answer: C

Solution:

Point O is typically used to denote the origin in a coordinate system.

(x, y, -z)

(x, -y, z)

(-x, y, z)

(x, y, z)

Correct Answer: A

Solution:

When a point is reflected across the XY-plane, the Z-coordinate changes sign while the X and Y coordinates remain the same. Thus, the new coordinates are

(x, y, -z)

To determine the color of the point

To locate the point in a plane

To locate the point in three-dimensional space

To calculate the distance from the origin

Correct Answer: C

Solution:

Three numbers are used to represent the coordinates of a point in space to locate it in three-dimensional space.

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: C

Solution:

In the diagram, the Z-axis is labeled as extending vertically.

(3, 2, 1)

(2, 3, 2)

(1, 2, 3)

(4, 1, 2)

Correct Answer: D

Solution:

Substitute the coordinates of each point into the plane equation

2x - 3y + 4z = 12

. For point

(4, 1, 2)

2(4) - 3(1) + 4(2) = 8 - 3 + 8 = 13.

This does not satisfy the equation. Re-evaluate the options. Correct point is

(1, 2, 3)

2(1) - 3(2) + 4(3) = 2 - 6 + 12 = 8.

None of the given options satisfy the equation, indicating an error in the options.

Horizontally to the left

Horizontally to the right

Vertically upward

Vertically downward

Correct Answer: B

Solution:

In a 3D coordinate system, the Y-axis extends horizontally to the right.

z = 3

x + y = 3

y = 2

x = 1

Correct Answer: A

Solution:

A plane parallel to the XY-plane has a constant

z

-coordinate. Therefore, the equation of the plane passing through

(1, 2, 3)

z = 3

XY-plane

XZ-plane

YZ-plane

None of these

Correct Answer: A

Solution:

The XY-plane is defined by the X and Y axes.

(2, 4, 6)

(1, 2, 3)

(3, 5, 7)

(1, 1, 1)

Correct Answer: B

Solution:

The direction vector of a line passing through two points

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

is given by

(x_2-x_1, y_2-y_1, z_2-z_1)

. Thus, the direction vector is

(3-1, 5-1, 7-1) = (2, 4, 6)

, which simplifies to

(1, 2, 3)

III

Correct Answer: D

Solution:

In the given coordinate system, octant IV is defined by

x > 0

y < 0

, and

z > 0

. Since point

P

has coordinates

(3, -2, 5)

, it satisfies these conditions and thus lies in the fourth octant.

(-2, -3, 5)

(2, 3, 5)

(2, -3, -5)

(-2, 3, -5)

Correct Answer: A

Solution:

Reflecting a point across the YZ-plane changes the sign of the x-coordinate. Thus, the new coordinates of

P

are

(-2, -3, 5)

Point O

Point A

Point C

Point D

Correct Answer: C

Solution:

Point C is on the top face of the prism as indicated in the diagram description.

Point P

Point O

Point A

Point N

Correct Answer: B

Solution:

The point O is typically labeled as the origin in a 3D coordinate system.

It requires one coordinate.

It requires two coordinates.

It requires three coordinates.

It requires four coordinates.

Correct Answer: C

Solution:

A point in 3D space is described by three coordinates, representing its distances from three mutually perpendicular planes.

(3, 4, -1)

(3, 4, 1)

(-3, -4, 1)

(1, -4, 3)

Correct Answer: A

Solution:

The normal vector to the plane

ax + by + cz = d

is given by

(a, b, c)

. Thus, for the plane

3x + 4y - z = 7

, the normal vector is

(3, 4, -1)

(1, 2, 3)

(-1, 2, 3)

(1, -2, 3)

(1, 2, -3)

Correct Answer: A

Solution:

A point lies in the first octant if all its coordinates are positive. Therefore, the point

(1, 2, 3)

lies in the first octant.

(2, 1, 2)

(3, 2, 1)

(4, 0, 2)

(1, 3, 2)

Correct Answer: A

Solution:

To verify which point lies on the plane, substitute the coordinates into the plane equation

3x - 4y + 5z = 20

. For point

(2, 1, 2)

3(2) - 4(1) + 5(2) = 6 - 4 + 10 = 12

. Since

12 \neq 20

, this point does not lie on the plane. Repeating for other points, only

(2, 1, 2)

satisfies the equation.

(3, 3, 3)

(1, 1, 1)

(2, 3, 4)

(3, 2, 1)

Correct Answer: A

Solution:

The direction vector of a line passing through two points

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

is given by

(x_2-x_1, y_2-y_1, z_2-z_1)

. Thus, the direction vector is

(4-1, 5-2, 6-3) = (3, 3, 3)

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: A

Solution:

In a 3D coordinate system, the X-axis is typically considered horizontal.

\sqrt{a^2 + b^2 + c^2}

a + b + c

\sqrt{a^2 + b^2}

\sqrt{b^2 + c^2}

Correct Answer: A

Solution:

The length of the diagonal in a rectangular prism from the origin

(0,0,0)

to the opposite vertex

(a,b,c)

is given by the distance formula in 3D:

\sqrt{a^2 + b^2 + c^2}

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: C

Solution:

The Z-axis is typically considered vertical in a 3D coordinate system.

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: C

Solution:

In a 3D coordinate system, the Z-axis is usually the one that extends vertically upward.

(3, 4, 6)

(3, 4, 5)

(2, 3, 4)

(3, 3, 3)

Correct Answer: A

Solution:

The direction vector of a line passing through two points

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

is given by

(x_2 - x_1, y_2 - y_1, z_2 - z_1)

. Thus, the direction vector is

(4 - 1, 6 - 2, 9 - 3) = (3, 4, 6)

4

-3

2

-2

Correct Answer: A

Solution:

The

x

coordinate represents the distance from the YZ-plane.

Correct Answer: C

Solution:

The equation of a sphere in standard form is

(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

, where

(h, k, l)

is the center and

r

is the radius. Here,

r^2 = 25

, so

r = 5

On the X-axis

On the Y-axis

On the Z-axis

On the XY-plane

Correct Answer: C

Solution:

The point

(0, 0, 5)

lies on the Z-axis as its

z

coordinate is non-zero while

x

and

y

are zero.

(1, 1, -2)

(1, 3, -8)

(7, 1, -2)

(1, 1, 3)

Correct Answer: A

Solution:

Translating a point

(x, y, z)

by a vector

(a, b, c)

results in a new point

(x+a, y+b, z+c)

. Therefore, the new coordinates of

R

are

(-3+4, 2-1, -5+3) = (1, 1, -2)

(10, 8, 5)

(10, 4, 5)

(8, 8, 5)

(8, 4, 5)

Correct Answer: A

Solution:

Substitute

t = 2

into the parametric equations:

x = 5(2) = 10

y = 2(2)^2 = 8

z = 2 + 3 = 5

. Therefore, the position of the airplane at

t = 2

(10, 8, 5)

120 cubic units

60 cubic units

30 cubic units

90 cubic units

Correct Answer: A

Solution:

The volume of a rectangular prism is given by the product of its length, width, and height. Here, the length is 4 units (from

O

A

), the width is 5 units (from

O

B

), and the height is 6 units (from

O

C

). Thus, the volume is

4 \times 5 \times 6 = 120

cubic units.

(2, -3, 4)

(4, -3, 2)

(2, 3, -4)

(1, -1, 1)

Correct Answer: A

Solution:

The normal vector to a plane given by the equation

ax + by + cz = d

(a, b, c)

. Therefore, the normal vector to the plane

2x - 3y + 4z = 12

(2, -3, 4)

Horizontally to the left

Horizontally to the right

Vertically upward

Vertically downward

Correct Answer: C

Solution:

In a 3D coordinate system, the Z-axis typically extends vertically upward.

Horizontally to the left

Horizontally to the right

Vertically upward

Vertically downward

Correct Answer: C

Solution:

The Z-axis extends vertically upward according to the diagram description.

Distance from the floor

Distance from two adjacent walls

Distance from the ceiling

Distance from a single wall

Correct Answer: D

Solution:

To locate a point in 3D space, we need the distances from three mutually perpendicular planes, typically the floor and two adjacent walls.

On the XZ-plane

On the XY-plane

On the YZ-plane

On all three planes

Correct Answer: A

Solution:

The point (0, 5, 0) lies on the XZ-plane because its y-coordinate is non-zero while the x and z coordinates are zero.

y = 3

x = 2

z = 4

x + y + z = 9

Correct Answer: A

Solution:

A plane parallel to the XZ-plane has a constant

y

-coordinate. Since the plane passes through the point

(2, 3, 4)

, the equation of the plane is

y = 3

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: C

Solution:

In a 3D coordinate system, the Z-axis is typically considered vertical.

(-1, -2, -3)

(1, -2, -3)

(-1, 2, -3)

(1, 2, 3)

Correct Answer: A

Solution:

The point (-1, -2, -3) has all negative coordinates, as each of the

x

y

, and

z

values are negative.

2 units

4 units

8 units

16 units

Correct Answer: B

Solution:

The equation of a sphere in 3D is given by

(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

, where

(a, b, c)

is the center and

r

is the radius. Comparing with

(x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 16

, we see

r^2 = 16

, so

r = \sqrt{16} = 4

units.

\sqrt{29}

\sqrt{25}

\sqrt{49}

\sqrt{41}

Correct Answer: A

Solution:

The distance from point

P(2, 3, 4)

to the origin

O(0, 0, 0)

can be calculated using the distance formula:

d = \sqrt{(2-0)^2 + (3-0)^2 + (4-0)^2} = \sqrt{4 + 9 + 16} = \sqrt{29}.

X-axis

Y-axis

Z-axis

None of the above

Correct Answer: A

Solution:

In the given 3D coordinate system, the X-axis is described as extending horizontally to the left.

(1, 2, 3)

(-1, 2, 3)

(1, -2, 3)

(1, 2, -3)

Correct Answer: A

Solution:

In the first octant, all coordinates (x, y, z) are positive.

Two numbers representing distances from two perpendicular lines

Three numbers representing distances from three perpendicular planes

One number representing distance from a single plane

Four numbers representing distances from four perpendicular planes

Correct Answer: B

Solution:

In 3D space, a point is located using three numbers representing distances from three mutually perpendicular planes.

(x+a, y+b, z+c)

(x-a, y-b, z-c)

(x+a, y-b, z+c)

(x-a, y+b, z-c)

Correct Answer: A

Solution:

Translation by a vector

\vec{v} = (a, b, c)

involves adding the vector components to the corresponding coordinates of the point. Thus, the new coordinates are

(x+a, y+b, z+c)

\sqrt{77}

\sqrt{61}

\sqrt{70}

\sqrt{50}

Correct Answer: A

Solution:

The diagonal of the rectangular prism can be calculated using the distance formula in 3D:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

. Substituting the coordinates of the opposite vertex

(4, 5, 6)

, we get

d = \sqrt{4^2 + 5^2 + 6^2} = \sqrt{16 + 25 + 36} = \sqrt{77}

X, Y, Z

Y, Z, X

Z, X, Y

X, Z, Y

Correct Answer: A

Solution:

In a 3D coordinate system, the axes are typically ordered as X, Y, and Z.

First octant

Second octant

Fourth octant

Fifth octant

Correct Answer: C

Solution:

The point

(3, -2, 5)

has a positive

x

-coordinate, a negative

y

-coordinate, and a positive

z

-coordinate, placing it in the fourth octant.

(1, 3, 2)

(7, -7, 4)

(1, 3, 4)

(7, 3, 2)

Correct Answer: A

Solution:

The translation of a point

(x, y, z)

by a vector

\vec{v} = (a, b, c)

results in a new point

(x+a, y+b, z+c)

. Applying this to point

P(4, -2, 3)

with vector

\vec{v} = (-3, 5, -1)

gives us the new coordinates:

(4-3, -2+5, 3-1) = (1, 3, 2)

True or False

Correct Answer: True

Solution:

In three-dimensional geometry, a point is located using three coordinates corresponding to its distances from three mutually perpendicular planes, typically the floor and two adjacent walls.

Correct Answer: False

Solution:

In a 3D coordinate system, the position of a point requires three numbers representing its perpendicular distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

The excerpt explains that a point in space has three coordinates representing its distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

The diagram description indicates that the diagonal runs from point P to Q across the rectangular prism.

Correct Answer: False

Solution:

In a three-dimensional Cartesian coordinate system, the Z-axis extends vertically, not horizontally.

Correct Answer: True

Solution:

A rectangular prism is a three-dimensional figure with right angles at its corners, as indicated by the angles at points N and A in the diagram.

Correct Answer: False

Solution:

In the provided diagram, the Y-axis extends horizontally to the right.

Correct Answer: False

Solution:

The Z-axis in a 3D coordinate system is typically oriented vertically, as described in the diagram.

Correct Answer: True

Solution:

The coordinates in 3D space are determined by the perpendicular distances from three mutually perpendicular planes, such as two walls and the floor.

Correct Answer: False

Solution:

In the provided excerpts, the X-axis extends horizontally to the left, not to the right.

Correct Answer: False

Solution:

In three-dimensional geometry, a point in space is defined by three coordinates, representing its position relative to three mutually perpendicular planes.

Correct Answer: True

Solution:

In three-dimensional space, a point is located by its distances from the three coordinate planes, which are perpendicular to each other.

Correct Answer: True

Solution:

A rectangular prism is defined by having right angles at its vertices, as indicated by the angles at points N and A.

Correct Answer: True

Solution:

In three-dimensional geometry, a point's position is defined by its distances from three mutually perpendicular planes, typically the floor and two adjacent walls in a room.

Correct Answer: False

Solution:

The Z-axis in the diagram is directed vertically upward, not horizontally.

Correct Answer: True

Solution:

The X-axis and Y-axis in a 3D coordinate system are mutually perpendicular, forming part of the Cartesian coordinate system.

Correct Answer: False

Solution:

In three-dimensional space, a point requires three coordinates representing distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

In three-dimensional geometry, a point is described by three coordinates representing its distances from three mutually perpendicular planes.

Correct Answer: False

Solution:

The diagram description specifies that the X-axis extends horizontally to the left, while the Y-axis extends horizontally to the right.

Correct Answer: False

Solution:

In the provided description, the Y-axis extends horizontally to the right, not vertically upward.

Correct Answer: False

Solution:

A point in space requires three coordinates, representing the perpendicular distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

The diagram description indicates that the X-axis extends horizontally to the left.

Correct Answer: True

Solution:

The diagonal of a rectangular prism runs across the prism, connecting two non-adjacent vertices, such as from P to Q.

Correct Answer: True

Solution:

According to the diagram description, the X-axis extends horizontally to the left and the Y-axis extends horizontally to the right.

Correct Answer: False

Solution:

In a 3D coordinate system, the X-axis is typically oriented horizontally.

Correct Answer: False

Solution:

Coordinates in a 3D space can be positive, negative, or zero, depending on their position relative to the origin.

Correct Answer: True

Solution:

In the provided diagram, the X-axis is oriented horizontally, extending from the origin to point A.

Correct Answer: True

Solution:

The diagram description explicitly states that the angles at points N and A are marked as 90°, indicating right angles.

Correct Answer: False

Solution:

The X-axis in a 3D coordinate system extends horizontally.

Correct Answer: False

Solution:

To locate a point in space, three coordinates are necessary, representing distances from three mutually perpendicular planes.

Correct Answer: False

Solution:

In a 3D coordinate system, the Z-axis extends vertically.

Correct Answer: True

Solution:

The coordinates of a point in space are determined by its perpendicular distances from two walls and the floor, which form the three coordinate planes.

Correct Answer: True

Solution:

Two intersecting mutually perpendicular lines in the plane are used to determine the coordinates of a point.

Correct Answer: False

Solution:

A rectangular prism is defined by having all right angles (90°) at its vertices.

Correct Answer: True

Solution:

The table of coordinates shows that points can have negative values for x, y, and z in certain octants.

Correct Answer: True

Solution:

In a 3D coordinate system, a point can be located in a quadrant where all its coordinates (x, y, z) are negative.

Correct Answer: True

Solution:

In a standard 3D coordinate system, the Z-axis is oriented vertically.

Correct Answer: False

Solution:

A point in space requires three coordinates, representing distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

The Y-axis is described as extending horizontally to the right in the diagram.

Correct Answer: False

Solution:

The coordinates of a point in space can be positive or negative, as shown in the coordinate table.

Correct Answer: False

Solution:

The X-axis and Y-axis in a 3D coordinate system extend horizontally but not necessarily in opposite directions. Their orientation depends on the specific setup of the coordinate system.

Correct Answer: True

Solution:

A point in space requires three numbers representing the perpendicular distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

The coordinates of a point in space are described as the perpendicular distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

In a three-dimensional coordinate system, the Z-axis is typically oriented vertically.

Correct Answer: False

Solution:

While the provided diagram shows right angles at certain points, a rectangular prism can theoretically have angles that are not right angles depending on the context.

Correct Answer: False

Solution:

In a standard 3D coordinate system, the Z-axis typically extends vertically upward, not downward.

Correct Answer: True

Solution:

A rectangular prism, by definition, has right angles at its vertices, as indicated in the description of the diagram.

Correct Answer: True

Solution:

In three-dimensional geometry, a point is located using three coordinates that represent its distances from three perpendicular planes: the floor and two walls.

Correct Answer: True

Solution:

The diagonal of a rectangular prism, such as from point P to Q, runs between two non-adjacent vertices.

Correct Answer: False

Solution:

To locate a point in a plane, only two intersecting mutually perpendicular lines, called coordinate axes, are required.

Correct Answer: True

Solution:

Coordinates in a 3D system can take any real value, including positive, negative, or zero, depending on the point's position relative to the origin.

Correct Answer: True

Solution:

To locate a point in 3D space, the distances from two perpendicular walls and the floor are used as coordinates.

Correct Answer: False

Solution:

The excerpt states that a point in space requires three coordinates, representing distances from three mutually perpendicular planes.

Correct Answer: True

Solution:

Coordinates in a 3D space can be positive or negative, depending on the point's position relative to the origin.

Correct Answer: True

Solution:

In the provided diagram description, the Z-axis is described as extending vertically upward, which is a common representation in a 3D Cartesian coordinate system.

Correct Answer: True

Solution:

The table of coordinates shows that x, y, and z values can be positive, negative, or zero, depending on the position of the point within the coordinate system.

Correct Answer: False

Solution:

In a 3D coordinate system, a point is determined by three coordinates representing distances from three mutually perpendicular planes.

Correct Answer: False

Solution:

According to the diagram description, the Z-axis extends vertically upward.

Correct Answer: False

Solution:

In a 3D coordinate system, the Z-axis is typically oriented vertically.

Correct Answer: False

Solution:

The provided diagram description indicates that the angles at points N and A of the rectangular prism are marked as 90°, indicating right angles.

Correct Answer: True

Solution:

In the first octant of a 3D coordinate system, all coordinate values (x, y, z) are positive.

Correct Answer: True

Solution:

The X-axis is described as extending horizontally to the left in the provided diagram description.

Introduction to Three Dimensional Geometry

AI Learning Assistant

Summary

Chapter 11: Introduction to Three Dimensional Geometry

Summary

Learning Objectives

Learning Objectives

Detailed Notes

Introduction to Three Dimensional Geometry

11.1 Introduction

11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space

11.3 Coordinates of a Point in Space

11.4 Distance between Two Points

Example Problems

Table of Octants

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips in Three Dimensional Geometry

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.Consider a rectangular prism in a 3D coordinate system with vertices at (0,0,0)(0,0,0)(0,0,0), (a,0,0)(a,0,0)(a,0,0), (0,b,0)(0,b,0)(0,b,0), and (0,0,c)(0,0,c)(0,0,c). What is the volume of the prism?

Correct Answer: A

Q2.If a point QQQ is located at (3,−4,5)(3, -4, 5)(3,−4,5) in a 3D space, what will be its coordinates after a translation by vector v⃗=(1,2,−3)\vec{v} = (1, 2, -3)v=(1,2,−3)?

Correct Answer: A

Q3.Consider a point PPP located at (4,−2,3)(4, -2, 3)(4,−2,3) in a 3D coordinate system. What will be the coordinates of PPP after it is reflected across the XY-plane?

Correct Answer: A

Q4.Which of the following transformations will reflect a point across the YZ-plane in a 3D coordinate system?

Correct Answer: A

Q5.Which of the following points has all positive coordinates?

Correct Answer: A

Q6.Consider a point P(x,y,z)P(x, y, z)P(x,y,z) in a 3D coordinate system. If PPP is located at (2,−3,4)(2, -3, 4)(2,−3,4), in which octant does this point lie?

Correct Answer: C

Q7.In a 3D coordinate system, a rectangular prism has vertices at (0,0,0)(0,0,0)(0,0,0), (4,0,0)(4,0,0)(4,0,0), (0,3,0)(0,3,0)(0,3,0), and (0,0,5)(0,0,5)(0,0,5). What is the volume of the prism?

Correct Answer: A

Q8.Which of the following transformations will rotate a point (x,y,z)(x, y, z)(x,y,z) 90 degrees counterclockwise about the Z-axis?

Correct Answer: A

Q9.Which of the following points lies on the Y-axis in a 3D coordinate system?

Correct Answer: B

Q10.In a 3D coordinate system, which of the following points has a negative xxx coordinate and positive yyy and zzz coordinates?

Correct Answer: A

Q11.If a point in space has coordinates (x,y,z)(x, y, z)(x,y,z) where x>0x > 0x>0, y<0y < 0y<0, and z>0z > 0z>0, in which octant does it lie?

Correct Answer: D

Q12.What is the volume of a rectangular prism with vertices at (0,0,0)(0,0,0)(0,0,0), (3,0,0)(3,0,0)(3,0,0), (0,4,0)(0,4,0)(0,4,0), and (0,0,6)(0,0,6)(0,0,6)?

Correct Answer: A

Q13.If a point in space has coordinates (3, -2, 5), which coordinate represents the height from the floor?

Correct Answer: C

Q14.Which of the following points is likely to be on the top face of a rectangular prism in a 3D coordinate system?

Correct Answer: C

Q15.In a 3D coordinate system, which of the following points has all positive coordinates?

Correct Answer: A

Q16.What is the primary purpose of using three numbers to represent the coordinates of a point in space?

Correct Answer: C

Q17.If a point in space has coordinates (0, 0, 0), what is its position relative to the coordinate planes?

Correct Answer: D

Q18.In the diagram of a rectangular prism, which angle is marked as 90°?

Correct Answer: C

Q19.In the coordinate system described, which point is likely to be at the origin?

Correct Answer: C

Q20.Consider a point PPP in a 3D coordinate system with coordinates (x,y,z)(x, y, z)(x,y,z). If PPP is reflected across the XY-plane, what will be the new coordinates of PPP?

Correct Answer: A

Q21.Which of the following best describes the purpose of using three numbers to represent the coordinates of a point in space?

Correct Answer: C

Q22.In the diagram of a rectangular prism within a 3D coordinate system, which axis is labeled as extending vertically?

Correct Answer: C

Q23.A plane is defined by the equation 2x−3y+4z=122x - 3y + 4z = 122x−3y+4z=12. Which of the following points lies on this plane?

Correct Answer: D

Q24.In a 3D coordinate system, which of the following describes the orientation of the Y-axis?

Correct Answer: B

Q25.Which of the following is the equation of a plane parallel to the XY-plane and passing through the point (1,2,3)(1, 2, 3)(1,2,3)?

Correct Answer: A

Q26.Which coordinate plane is defined by the X and Y axes?

Correct Answer: A

Q27.A line passes through points (1,1,1)(1, 1, 1)(1,1,1) and (3,5,7)(3, 5, 7)(3,5,7) in 3D space. What is the direction vector of this line?

Correct Answer: B

Q28.Consider a point P(x,y,z)P(x, y, z)P(x,y,z) located inside a rectangular prism. If the coordinates of point PPP are (3,−2,5)(3, -2, 5)(3,−2,5), which octant does this point lie in according to the given coordinate system?

Correct Answer: D

Q29.In a 3D coordinate system, if a point PPP is located at (2,−3,5)(2, -3, 5)(2,−3,5), what will be the coordinates of PPP after it is reflected across the YZ-plane?

Correct Answer: A

Q30.In the diagram of a rectangular prism within a 3D coordinate system, which point is likely to be at the top face of the prism?