In the three-dimensional coordinate system, if a point $P(x, y, z)$ is equidistant from the points $A(1, 2, 3)$ and $B(4, 5, 6)$, what is the equation representing this condition?

$(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = (x - 4)^2 + (y - 5)^2 + (z - 6)^2$

$(x + 1)^2 + (y + 2)^2 + (z + 3)^2 = (x + 4)^2 + (y + 5)^2 + (z + 6)^2$

$(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = (x + 4)^2 + (y + 5)^2 + (z + 6)^2$

$(x + 1)^2 + (y - 2)^2 + (z - 3)^2 = (x - 4)^2 + (y - 5)^2 + (z - 6)^2$

Given points $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$ in three-dimensional space, determine if these points are collinear.

Yes, because the vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are parallel.

No, because the vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are not parallel.

Yes, because the distance $AB + BC = AC$.

No, because the distance $AB + BC \neq AC$.

What is the significance of the origin in a three-dimensional coordinate system?

It is the point where the x, y, and z axes intersect.

It is the point where the x and y axes intersect.

It is the point where the y and z axes intersect.

It is the point where the x and z axes intersect.

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Introduction to Three Dimensional Geometry

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

3D Distance-Based Shape Verification

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Difficulty Level:

High

Centroid and Missing Coordinate Problems

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Collinearity of Points in 3D

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Learning Objectives

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Learning Objectives

Understand the structure and components of the Three Dimensional Coordinate System, including coordinate axes, planes, and octants.
Calculate the Coordinates of a Point in Space using perpendicular distances from the coordinate planes.
Derive and apply the Distance Formula in 3D Space to find the distance between two points.
Determine the Collinearity of Points in 3D using the distance formula.
Formulate the Equation of a Set of Points equidistant from two given points in three-dimensional space.
Identify the Octants and Sign of Coordinates by analyzing the signs of $(x, y, z)$ coordinates.
Recognize the forms of Points on Axes and Coordinate Planes, such as $(x,0,0)$ , $(0,y,0)$ , and $(0,0,z)$ .
Verify geometric shapes like isosceles/right triangles and parallelograms using 3D Distance-Based Shape Verification.
Solve problems involving the Centroid and Missing Coordinate Problems by using centroid relations and coordinate geometry in 3D.

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Revision Notes & Summary

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Chapter Notes

Three Dimensional Coordinate System

Coordinate Axes and Planes: In three-dimensional space, three planes intersect at a point called the origin ( $O$ ), forming the $x$ , $y$ , and $z$ axes. These axes are mutually perpendicular.
Octants: The space is divided into eight parts known as octants, denoted as $XOYZ$ , $X'OYZ$ , $X'OY'Z$ , $XOY'Z$ , $XOYZ'$ , $X'OYZ'$ , $X'OY'Z'$ , and $XOY'Z'$ .

Coordinates of a Point in Space

Definition: A point $P$ in space is represented by an ordered triplet $(x, y, z)$ , which are the perpendicular distances from the $YZ$ , $ZX$ , and $XY$ planes, respectively.
Location: To locate a point $(x, y, z)$ , draw perpendiculars from $P$ to the coordinate planes, intersecting the axes at points that determine the coordinates.

Distance Formula in 3D Space

Formula: The distance $PQ$ between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Collinearity of Points in 3D

Condition: Points $P$ , $Q$ , and $R$ are collinear if the sum of the distances $PQ$ and $QR$ equals the distance $PR$ .

Equation of a Set of Points

Equidistant Points: The equation of a set of points equidistant from two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is derived from the condition $PA = PB$ .

Octants and Sign of Coordinates

Sign Table:
Octant $x$ $y$ $z$
I + + +
II - + +
III - - +
IV + - +
V + + -
VI - + -
VII - - -
VIII + - -

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

Points on Axes and Coordinate Planes

Forms:
- On $x$ -axis: $(x, 0, 0)$
- On $y$ -axis: $(0, y, 0)$
- On $z$ -axis: $(0, 0, z)$
- In $XY$ -plane: $(x, y, 0)$
- In $YZ$ -plane: $(0, y, z)$
- In $ZX$ -plane: $(x, 0, z)$

3D Distance-Based Shape Verification

Verification: Use the distance formula to verify geometric properties such as isosceles or right triangles, parallelograms, and rectangles.

Centroid and Missing Coordinate Problems

Centroid Formula: For a triangle with vertices $A(x_1, y_1, z_1)$ , $B(x_2, y_2, z_2)$ , $C(x_3, y_3, z_3)$ , the centroid $G$ is given by: $G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)$
Finding Missing Coordinates: Use the centroid formula to solve for missing coordinates when the centroid is known.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

\sqrt{27}

\sqrt{35}

\sqrt{29}

\sqrt{45}

Correct Answer: C

Solution:

The distance between two points

(x_1, y_1, z_1)

and

(x_2, y_2, z_2)

is given by

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

. Substituting the given points, we get

\sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{3^2 + 3^2 + 3^2} = \sqrt{27} = \sqrt{29}

Chapter Concept:

Three Dimensional Coordinate System

The x-coordinate is zero.

The y-coordinate is zero.

The z-coordinate is zero.

None of the above.

Correct Answer: A

Solution:

A point on the YZ-plane has coordinates of the form

(0, y, z)

, meaning the x-coordinate is zero.

Chapter Concept:

Distance Formula in 3D Space

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True or False

Correct Answer: True

Solution:

In three-dimensional geometry, a point on the x-axis is represented by

(x, 0, 0)

, indicating that it lies on the x-axis with no displacement along the y or z axes.

Chapter Concept :

Points on Axes and Coordinate Planes

Correct Answer: False

Solution:

A point with coordinates

(-3, 1, 2)

lies in the second octant, as indicated by the signs of the coordinates:

x

is negative,

y

is positive, and

z

is positive.

Chapter Concept :

Octants and Sign of Coordinates

Correct Answer: True

Solution:

The three planes intersect along the lines X'OX, Y'OY, and Z'OZ, which are the x, y, and z-axes, respectively, and are mutually perpendicular.

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Introduction to Three Dimensional Geometry

Study Action Plan (Sorted by Difficulty)

3D Distance-Based Shape Verification

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Centroid and Missing Coordinate Problems

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Collinearity of Points in 3D

Learning Objectives

Learning Objectives

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Revision Notes & Summary

Chapter Notes

Three Dimensional Coordinate System

Coordinates of a Point in Space

Distance Formula in 3D Space

Collinearity of Points in 3D

Equation of a Set of Points

Octants and Sign of Coordinates

Points on Axes and Coordinate Planes

3D Distance-Based Shape Verification

Centroid and Missing Coordinate Problems

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Given the points A(1,2,3)A(1, 2, 3)A(1,2,3) and B(4,5,6)B(4, 5, 6)B(4,5,6) in a three-dimensional space, what is the distance between these two points?

Correct Answer: C

Q2.Which of the following statements is true about the coordinates of a point on the YZ-plane?

Correct Answer: A

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True or False

Q1.A point on the x-axis in three-dimensional space has coordinates of the form (x,0,0)(x, 0, 0)(x,0,0). This means its y and z coordinates are zero.

Correct Answer: True

Q2.In the three-dimensional coordinate system, a point with coordinates (−3,1,2)(-3, 1, 2)(−3,1,2) lies in the first octant.

Correct Answer: False

Q3.In a three-dimensional coordinate system, the three planes intersect along lines that are mutually perpendicular to each other.

Correct Answer: True

High

Octants and Sign of Coordinates

High

Three Dimensional Coordinate System

Low

Coordinates of a Point in Space

Low

Distance Formula in 3D Space

Low

Equation of a Set of Points

Low

Points on Axes and Coordinate Planes

Low

Q3.Which of the following points lies in the first octant of a three-dimensional coordinate system?

Correct Answer: D

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Q4.Which of the following points lies in the second octant?

Correct Answer: B

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Q5.Determine the octant in which the point (−3,−4,5)(-3, -4, 5)(−3,−4,5) lies in a 3D coordinate system.

Correct Answer: C

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Q6.In the three-dimensional coordinate system, if a point P(x,y,z)P(x, y, z)P(x,y,z) is equidistant from the points A(1,2,3)A(1, 2, 3)A(1,2,3) and B(4,5,6)B(4, 5, 6)B(4,5,6), what is the equation representing this condition?

Correct Answer: A

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Q7.Given points A(1,2,3)A(1, 2, 3)A(1,2,3), B(4,5,6)B(4, 5, 6)B(4,5,6), and C(7,8,9)C(7, 8, 9)C(7,8,9) in three-dimensional space, determine if these points are collinear.

Correct Answer: A

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Q8.What are the coordinates of the origin in a three-dimensional coordinate system?

Correct Answer: A

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Q9.What is the significance of the origin in a three-dimensional coordinate system?

Correct Answer: A