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.

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.

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.

Introduction to Three Dimensional Geometry

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

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.

CBSE Revision Notes & Quick Summary for Last-Minute Study

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	+	+	+	+	-	-	-	-

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

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.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

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.

(-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)

(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)

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.

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.

Correct Answer: C

Solution:

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

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.

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.

(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)

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.

4

-3

2

-2

Correct Answer: A

Solution:

The

x

coordinate represents the distance from the YZ-plane.

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.

(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)

(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.

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.

(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)

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.

XY-plane

XZ-plane

YZ-plane

None of these

Correct Answer: A

Solution:

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

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.

(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)

(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)

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.

(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.

\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}.

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, 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.

\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.

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.

(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)

-2

None of these

Correct Answer: C

Solution:

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

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.

(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.

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.

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.

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.

(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.

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.

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.

(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)

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 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.

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

(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)

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.

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.

(-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.

(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)

\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}

Correct Answer: D

Solution:

In the fifth octant, the coordinates are

x > 0

y < 0

, and

z > 0

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.

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.

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.

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

(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)

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: A

Solution:

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

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.

(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.

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

(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)

(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)

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

Solution:

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

Introduction to Three Dimensional Geometry

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

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

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips in Three Dimensional Geometry

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.In the diagram of a rectangular prism within a 3D coordinate system, which axis is labeled as extending vertically?

Correct Answer: C

Q2.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

Q3.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

Q4.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

Q5.Which of the following best describes the position of a point in 3D space?

Correct Answer: C

Q6.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

Q7.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

Q8.A rectangular prism has its vertices at O(0,0,0)O(0,0,0)O(0,0,0), A(4,0,0)A(4,0,0)A(4,0,0), B(0,5,0)B(0,5,0)B(0,5,0), and C(0,0,6)C(0,0,6)C(0,0,6). What is the volume of this prism?

Correct Answer: A

Q9.What is the orientation of the Z-axis in the provided 3D coordinate system?

Correct Answer: C

Q10.A plane in 3D space is given by the equation 3x+4y−z=73x + 4y - z = 73x+4y−z=7. What is the normal vector to this plane?

Correct Answer: A

Q11.In the diagram of a rectangular prism within a 3D coordinate system, which point is likely to be at the origin?

Correct Answer: B

Q12.If a point in space has coordinates (4,−3,2)(4, -3, 2)(4,−3,2), which coordinate represents the distance from the YZ-plane?

Correct Answer: A

Q13.In the coordinate system described, which point is likely to be at the origin?

Correct Answer: C

Q14.If a line passes through the points (1,2,3)(1, 2, 3)(1,2,3) and (4,6,9)(4, 6, 9)(4,6,9), what is the direction vector of this line?

Correct Answer: A

Q15.Which of the following points lies in the first octant of a 3D coordinate system?

Correct Answer: A

Q16.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

Q17.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

Q18.If a point in space has coordinates (0, 0, 0), what is its position relative to the coordinate planes?

Correct Answer: D

Q19.Which coordinate plane is defined by the X and Y axes?

Correct Answer: A

Q20.Which of the following points has all positive coordinates?

Correct Answer: A

Q21.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

Q22.Consider a point PPP located at (4,−2,3)(4, -2, 3)(4,−2,3) in a 3D coordinate system. If PPP is translated by the vector v⃗=(−3,5,−1)\vec{v} = (-3, 5, -1)v=(−3,5,−1), what are the new coordinates of PPP?

Correct Answer: A

Q23.Which of the following transformations will reflect a point across the YZ-plane in a 3D coordinate system?

Correct Answer: A

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

Correct Answer: C

Q25.Which of the following points lies on the Y-axis in a 3D coordinate system?

Correct Answer: B

Correct Answer: A

Q27.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

Q28.In a 3D coordinate system, a plane is defined by the equation 3x−4y+5z=203x - 4y + 5z = 203x−4y+5z=20. Which of the following points lies on this plane?

Correct Answer: A

Q29.A rectangular prism is positioned in a 3D coordinate system with its vertices at O(0,0,0)O(0, 0, 0)O(0,0,0), A(4,0,0)A(4, 0, 0)A(4,0,0), B(0,5,0)B(0, 5, 0)B(0,5,0), and C(0,0,6)C(0, 0, 6)C(0,0,6). What is the length of the diagonal from the origin OOO to the opposite vertex of the prism?

Correct Answer: A

Q30.In a 3D coordinate system, which of the following represents the correct order of axes?

Correct Answer: A

Q31.In a 3D coordinate system, which of the following axes typically extends horizontally to the left?

Correct Answer: A

Q32.In a 3D coordinate system, if a line passes through points (1,2,3)(1, 2, 3)(1,2,3) and (4,5,6)(4, 5, 6)(4,5,6), what is the direction vector of the line?

Correct Answer: A