Question 1

If the points $P(-2, 3, 5)$, $Q(1, 2, 3)$, and $R(7, 0, -1)$ are given, what is the relationship between the distances $PQ$, $QR$, and $PR$?

Accepted Answer

Correct Option: A. Solution: The points $P$, $Q$, and $R$ are collinear because the sum of the distances $PQ$ and $QR$ equals the distance $PR$.

Question 2

A point $P(x, y, z)$ lies in the octant where $x > 0$, $y < 0$, and $z > 0$. Identify the octant in which the point lies.

Accepted Answer

Correct Option: C. Solution: In the fourth octant, the signs of the coordinates are $x > 0$, $y < 0$, and $z > 0$.

Question 3

What is the equation of the set of points $P(x, y, z)$ such that the sum of the distances from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to 10?

Accepted Answer

Correct Option: A. Solution: The condition $PA + PB = 10$ where $PA = \sqrt{(x-4)^2 + y^2 + z^2}$ and $PB = \sqrt{(x+4)^2 + y^2 + z^2}$ leads to the equation $x^2 + y^2 + z^2 = 25$ after simplification.

Question 4

Consider a point $P(x, y, z)$ in the three-dimensional coordinate system. If $P$ lies in the octant where $x > 0$, $y < 0$, and $z < 0$, which octant is $P$ located in?

Accepted Answer

Correct Option: C. Solution: In the VII octant, the signs of the coordinates are $x > 0$, $y < 0$, and $z < 0$, matching the conditions given for point $P$.

Question 5

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

Accepted Answer

Correct Option: A. Solution: A point on the YZ-plane has coordinates of the form $(0, y, z)$, meaning the x-coordinate is zero.

Question 6

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.

Accepted Answer

Correct Option: A. Solution: The vectors $\overrightarrow{AB} = (3, 3, 3)$ and $\overrightarrow{BC} = (3, 3, 3)$ are parallel, hence points $A$, $B$, and $C$ are collinear.

Question 7

Which of the following conditions must be satisfied for three points $P(x_1, y_1, z_1)$, $Q(x_2, y_2, z_2)$, and $R(x_3, y_3, z_3)$ to be collinear in three-dimensional space?

Accepted Answer

Correct Option: A. Solution: For three points to be collinear, the sum of the distances $PQ$ and $QR$ must equal the distance $PR$.

Question 8

Given the point $(4, -2, 3)$, identify the octant it lies in.

Accepted Answer

Correct Option: B. Solution: The point $(4, -2, 3)$ lies in the IV octant where $x$ is positive, $y$ is negative, and $z$ is positive.

Question 9

If a point $P(x, y, z)$ is equidistant from the x-axis, y-axis, and z-axis, which of the following must be true?

Accepted Answer

Correct Option: A. Solution: For a point to be equidistant from the x-axis, y-axis, and z-axis, all three coordinates must be equal, i.e., $x = y = z$.

Question 10

Given a point $P(x, y, z)$ lies in the YZ-plane, what can be inferred about its x-coordinate?

Accepted Answer

Correct Option: B. Solution: Points in the YZ-plane have coordinates of the form $(0, y, z)$, indicating that the x-coordinate must be zero.

Question 11

Consider a point $R(0, -5, 7)$ in three-dimensional space. Which plane does this point lie on?

Accepted Answer

Correct Option: A. Solution: The point $R(0, -5, 7)$ lies on the XZ-plane because its $y$-coordinate is zero.

Question 12

A point $S(-2, -3, -4)$ is in the seventh octant. What is the sign of each coordinate in this octant?

Accepted Answer

Correct Option: C. Solution: In the seventh octant, all coordinates are negative: $x < 0$, $y < 0$, $z < 0$.

Question 13

Find the coordinates of the fourth vertex $D$ of a parallelogram $ABCD$ given $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$.

Accepted Answer

Correct Option: A. Solution: In a parallelogram, opposite sides are equal and parallel. Using vector addition, $D = A + C - B = (1, 2, 3) + (7, 8, 9) - (4, 5, 6) = (10, 11, 12)$.

Question 14

Given a point $P(x, y, z)$ in space, which of the following statements is true if $P$ lies on the x-axis?

Accepted Answer

Correct Option: A. Solution: A point on the x-axis has coordinates of the form $(x, 0, 0)$, meaning both the y and z coordinates are zero.

Question 15

If a point has coordinates $(x, y, z)$ with $x > 0$, $y < 0$, and $z < 0$, which octant does it belong to?

Accepted Answer

Correct Option: D. Solution: The point belongs to the VIII octant where $x$ is positive, $y$ is negative, and $z$ is negative.

Question 16

Calculate the distance between the points $P(1, -3, 4)$ and $Q(-4, 1, 2)$ in three-dimensional space.

Accepted Answer

Correct Option: A. Solution: Using the distance formula: $PQ = \sqrt{(-4-1)^2 + (1+3)^2 + (2-4)^2} = \sqrt{25 + 16 + 4} = \sqrt{45}$.

Question 17

Given two points $A(2, -3, 4)$ and $B(-1, 2, -5)$ in a 3D space, what is the distance between them?

Accepted Answer

Correct Option: B. Solution: The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by the formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. Substituting the given values: $\sqrt{((-1) - 2)^2 + (2 - (-3))^2 + ((-5) - 4)^2} = \sqrt{(-3)^2 + 5^2 + (-9)^2} = \sqrt{9 + 25 + 81} = \sqrt{115}$.

Question 18

Given two points $P(1, -2, 3)$ and $Q(4, 2, -1)$ in a 3D space, what is the distance between them?

Accepted Answer

Correct Option: B. Solution: The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$. Substituting the given points: $\sqrt{(4 - 1)^2 + (2 + 2)^2 + (-1 - 3)^2} = \sqrt{3^2 + 4^2 + (-4)^2} = \sqrt{9 + 16 + 16} = \sqrt{41}$.

Question 19

If points $P(2, -1, 3)$, $Q(5, 2, 6)$, and $R(8, 5, 9)$ are given, find the condition for these points to be collinear.

Accepted Answer

Correct Option: B. Solution: For collinearity, the cross product of $\overrightarrow{PQ}$ and $\overrightarrow{QR}$ must be zero, indicating parallel vectors.

Question 20

If a point has coordinates $(x, y, z)$, what do these coordinates represent?

Accepted Answer

Correct Option: B. Solution: In three-dimensional space, the coordinates $(x, y, z)$ represent the perpendicular distances from the YZ, ZX, and XY planes respectively.

Question 21

In a three-dimensional coordinate system, if a point $P(a, b, c)$ lies on the $XZ$-plane, what can be said about its $y$-coordinate?

Accepted Answer

Correct Option: C. Solution: A point lying on the $XZ$-plane has its $y$-coordinate equal to zero.

Question 22

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

Accepted Answer

Correct Option: A. Solution: The origin is the point $(0, 0, 0)$ where the x, y, and z axes intersect in a three-dimensional coordinate system.

Question 23

Consider the points $A(-1, 0, 1)$, $B(2, 3, 4)$, and $C(5, 6, 7)$. Which of the following statements is true regarding their collinearity?

Accepted Answer

Correct Option: A. Solution: The vector $\overrightarrow{AB} = (3, 3, 3)$ is a scalar multiple of $\overrightarrow{BC} = (3, 3, 3)$, confirming collinearity.

Question 24

Which of the following points is equidistant from the origin and the point $(4, 0, 0)$?

Accepted Answer

Correct Option: A. Solution: The distance from the origin to $(4, 0, 0)$ is $4$. A point equidistant from both the origin and $(4, 0, 0)$ along the x-axis is $(2, 0, 0)$, since $\sqrt{(2-0)^2 + 0^2 + 0^2} = 2$ and $\sqrt{(4-2)^2 + 0^2 + 0^2} = 2$.

Question 25

Consider a point $P(x, y, z)$ in a 3D space. If $P$ is equidistant from the origin $(0, 0, 0)$ and the point $(4, 4, 4)$, which of the following could be the coordinates of $P$?

Accepted Answer

Correct Option: A. Solution: For $P(x, y, z)$ to be equidistant from the origin and the point $(4, 4, 4)$, the distances $\sqrt{x^2 + y^2 + z^2}$ and $\sqrt{(x-4)^2 + (y-4)^2 + (z-4)^2}$ must be equal. Solving this gives $x^2 + y^2 + z^2 = (x-4)^2 + (y-4)^2 + (z-4)^2$. Simplifying, we find $x = y = z = 2$. Therefore, the coordinates of $P$ could be $(2, 2, 2)$.

Question 26

Given three points $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$ in 3D space, determine if they form an isosceles triangle using the 3D distance formula.

Accepted Answer

Correct Option: B. Solution: To determine if the points form an isosceles triangle, calculate the distances between each pair of points using the distance formula: $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$. The distances are not equal, hence they do not form an isosceles triangle.

Question 27

In which octant does the point (-3, 1, 2) lie?

Accepted Answer

Correct Option: B. Solution: The point (-3, 1, 2) lies in the second octant where x is negative, y is positive, and z is positive.

Question 28

In which octant does the point $(-3, 1, -2)$ lie?

Accepted Answer

Correct Option: C. Solution: The point $(-3, 1, -2)$ has negative x and z coordinates and a positive y coordinate, placing it in the sixth octant.

Question 29

In a triangle with vertices $A(2, -1, 4)$, $B(3, 5, -2)$, and $C(x, y, z)$, the centroid is at $(3, 3, 2)$. What is the value of $z$?

Accepted Answer

Correct Option: B. Solution: The centroid $G$ of a triangle with vertices $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$ is given by $G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)$. Given $G = (3, 3, 2)$ and $A = (2, -1, 4)$, $B = (3, 5, -2)$, we solve for $C(x_3, y_3, z_3)$: $\frac{2 + 3 + x_3}{3} = 3$, $\frac{-1 + 5 + y_3}{3} = 3$, $\frac{4 - 2 + z_3}{3} = 2$. Solving these gives $x_3 = 4$, $y_3 = 1$, $z_3 = 2$. Therefore, the value of $z$ is 2.

Question 30

If a point is located on the x-axis in a three-dimensional coordinate system, what are its y-coordinate and z-coordinate?

Accepted Answer

Correct Option: A. Solution: A point on the x-axis has coordinates of the form $(x, 0, 0)$, meaning both y-coordinate and z-coordinate are zero.

Question 31

If the points $P(2, 3, 5)$, $Q(6, 7, 8)$, and $R(2, 3, 5)$ are vertices of a triangle in 3D space, what type of triangle is it?

Accepted Answer

Correct Option: D. Solution: The points $P$ and $R$ are identical, making the triangle degenerate as it collapses into a line segment.

Question 32

If the centroid of triangle $ABC$ is at $(1, 1, 1)$ and the coordinates of $A$ and $B$ are $(3, -5, 7)$ and $(-1, 7, -6)$ respectively, find the coordinates of point $C$.

Accepted Answer

Correct Option: A. Solution: Using the centroid formula $\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right) = (1, 1, 1)$, we solve for $C$ as $(1, 1, 2)$.

Question 33

Determine the equation of the set of points $P(x, y, z)$ such that $PA^2 + PB^2 = 2k^2$, where $A(3, 4, 5)$ and $B(-1, 3, -7)$.

Accepted Answer

Correct Option: A. Solution: By expanding the condition $PA^2 + PB^2 = 2k^2$, we derive the equation $2x^2 + 2y^2 + 2z^2 - 4x - 14y + 4z = 2k^2 - 109$.

Question 34

In a three-dimensional coordinate system, what are the coordinates of a point that lies on the YZ-plane?

Accepted Answer

Correct Option: C. Solution: A point on the YZ-plane has an x-coordinate of zero, hence its coordinates are $(0, y, z)$. This is because the YZ-plane is defined by the x-coordinate being zero.

Question 35

If a point $P(a, b, c)$ is located such that $a > 0$, $b < 0$, and $c > 0$, in which octant does it lie?

Accepted Answer

Correct Option: B. Solution: The point $P(a, b, c)$ with $a > 0$, $b < 0$, and $c > 0$ lies in the fourth octant.

Question 36

If a point $P(x, y, z)$ has coordinates $(0, 0, z)$, which axis does it lie on?

Accepted Answer

Correct Option: C. Solution: A point with coordinates $(0, 0, z)$ lies on the Z-axis, as the x and y coordinates are zero.

Question 37

Find the equation of the set of points equidistant from the points $A(1, 2, 3)$ and $B(3, 2, -1)$ in three-dimensional space.

Accepted Answer

Correct Option: A. Solution: The equation of the set of points equidistant from $A(1, 2, 3)$ and $B(3, 2, -1)$ is derived by setting the distances equal: $\sqrt{(x-1)^2 + (y-2)^2 + (z-3)^2} = \sqrt{(x-3)^2 + (y-2)^2 + (z+1)^2}$. Simplifying gives $x - 1 = 0$.

Question 38

A point is located on the z-axis in a three-dimensional coordinate system. Which of the following correctly represents the coordinates of this point?

Accepted Answer

Correct Option: C. Solution: Any point on the z-axis has coordinates of the form $(0, 0, z)$, where $z$ is the distance from the origin along the z-axis.

Question 39

What is the equation of the set of points equidistant from $(1, 2, 3)$ and $(3, 2, -1)$?

Accepted Answer

Correct Option: C. Solution: The equation of the set of points equidistant from $(1, 2, 3)$ and $(3, 2, -1)$ is $x = 2$.

Question 40

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?

Accepted Answer

Correct Option: A. Solution: The condition for a point $P(x, y, z)$ to be equidistant from two points $A(1, 2, 3)$ and $B(4, 5, 6)$ is given by the equation of equal distances: $(x - 1)^2 + (y - 2)^2 + (z - 3)^2 = (x - 4)^2 + (y - 5)^2 + (z - 6)^2$.

Question 41

What is the form of the coordinates of any point on the x-axis in three-dimensional space?

Accepted Answer

Correct Option: A. Solution: Any point on the x-axis has coordinates of the form (x, 0, 0) because the y and z coordinates are zero.

Question 42

If a point $Q(-3, 4, -2)$ is located in a certain octant, what are the signs of the coordinates in that octant?

Accepted Answer

Correct Option: A. Solution: The point $Q(-3, 4, -2)$ has coordinates with signs $x < 0$, $y > 0$, $z < 0$, which corresponds to the sixth octant.

Question 43

If a point $P(x, y, z)$ lies in the YZ-plane, what can be said about its x-coordinate?

Accepted Answer

Correct Option: A. Solution: A point in the YZ-plane has an x-coordinate of zero.

Question 44

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

Accepted Answer

Correct Option: 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 $\sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{3^2 + 3^2 + 3^2} = \sqrt{27} = \sqrt{29}$.

Question 45

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

Accepted Answer

Correct Option: D. Solution: The first octant is defined by all coordinates being positive. Therefore, the point $(3, 2, 1)$ lies in the first octant.

Question 46

If point $P(2, 3, 5)$ is in the first octant, which of the following points also lies in the same octant?

Accepted Answer

Correct Option: D. Solution: In the first octant, all coordinates $(x, y, z)$ are positive. The point $(3, 4, 6)$ has all positive coordinates, hence it lies in the first octant.

Question 47

What are the coordinates of a point on the x-axis in three-dimensional space?

Accepted Answer

Correct Option: A. Solution: Any point on the x-axis has coordinates of the form $(x, 0, 0)$, where $x$ is the distance from the origin along the x-axis.

Question 48

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

Accepted Answer

Correct Option: C. Solution: The signs of the coordinates are $x: -$, $y: -$, $z: +$. According to the octant sign table, this corresponds to the VII octant.

Question 49

Which of the following points lies in the second octant?

Accepted Answer

Correct Option: B. Solution: The point $(-3, 1, 2)$ lies in the second octant where $x$ is negative, $y$ is positive, and $z$ is positive.

Question 50

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

Accepted Answer

Correct Option: A. Solution: The origin in a three-dimensional coordinate system is represented by the coordinates (0, 0, 0).

Question 51

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

Accepted Answer

Correct Option: A. Solution: The points $A$, $B$, and $C$ are collinear because the sum of the distances $AB$ and $BC$ is equal to the distance $AC$.

Question 52

Given the points $A(1, 2, 3)$, $B(-1, -2, -1)$, $C(2, 3, 2)$, and $D(4, 7, 6)$, which of the following statements is true about the quadrilateral $ABCD$?

Accepted Answer

Correct Option: B. Solution: To show $ABCD$ is a parallelogram, we need to show opposite sides are equal. Calculating the distances, $AB = CD$ and $BC = DA$, confirming it is a parallelogram. However, the diagonals $AC$ and $BD$ are not equal, so it is not a rectangle.

Question 53

A point $P(x, y, z)$ is reflected across the XY-plane. What are the coordinates of the reflected point?

Accepted Answer

Correct Option: A. Solution: Reflecting a point across the XY-plane changes the sign of the z-coordinate. Thus, the reflected point is $(x, y, -z)$.

Question 54

The centroid of a triangle is at the point $(1, 1, 1)$. If the coordinates of vertices $A$ and $B$ are $(3, -5, 7)$ and $(-1, 7, -6)$ respectively, what are the coordinates of vertex $C$?

Accepted Answer

Correct Option: A. Solution: Using the centroid formula $G(x, y, z) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)$, substitute $G(1, 1, 1)$, $A(3, -5, 7)$, $B(-1, 7, -6)$ to find $C(1, 1, 2)$.

Question 55

For points $P(1, 2, 3)$, $Q(4, 5, 6)$, and $R(7, 8, 9)$ in space, determine the condition for collinearity using the distance formula.

Accepted Answer

Correct Option: A. Solution: For collinearity, the sum of the distances $PQ + QR$ must equal $PR$, which holds true for these points.

Question 56

Consider a point $P(x, y, z)$ in space. If $P$ is located in the fourth octant, what can be said about the signs of its coordinates?

Accepted Answer

Correct Option: C. Solution: In the fourth octant, the x-coordinate is positive, the y-coordinate is negative, and the z-coordinate is negative.

Question 57

If a point P has coordinates (x, y, z), what do these coordinates represent in three-dimensional space?

Accepted Answer

Correct Option: B. Solution: In three-dimensional space, the coordinates (x, y, z) represent the perpendicular distances from the YZ, ZX, and XY planes, respectively.

Question 58

Given a triangle with vertices $A(3, -5, 7)$, $B(-1, 7, -6)$, and $C(x, y, z)$, if the centroid is $(1, 1, 1)$, what is the value of $z$?

Accepted Answer

Correct Option: A. Solution: Using the centroid formula for the $z$-coordinate: $1 = \frac{7 - 6 + z}{3}$, solving gives $z = 2$.

Question 59

The distance formula in three-dimensional space between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Accepted Answer

Answer: True. Solution: The distance formula in three-dimensional space is derived from the Pythagorean theorem and calculates the Euclidean distance between two points.

Question 60

In the three-dimensional coordinate system, the coordinates of any point on the y-axis are of the form $(0, y, 0)$.

Accepted Answer

Answer: True. Solution: A point on the y-axis has no x or z displacement, hence its coordinates are $(0, y, 0)$.

Question 61

In three-dimensional space, the coordinates of a point are determined by its perpendicular distances from the x, y, and z-axes.

Accepted Answer

Answer: False. Solution: The coordinates of a point in three-dimensional space are determined by its perpendicular distances from the YZ, ZX, and XY planes, not directly from the axes.

Question 62

The coordinates of a point on the x-axis in three-dimensional space can be represented as $(x, y, z)$ where $y$ and $z$ are non-zero.

Accepted Answer

Answer: False. Solution: A point on the x-axis has coordinates of the form $(x, 0, 0)$, meaning both $y$ and $z$ are zero.

Question 63

In a three-dimensional Cartesian coordinate system, the signs of the coordinates determine the octant in which a point lies.

Accepted Answer

Answer: True. Solution: The signs of the coordinates $(x, y, z)$ determine the octant in which the point is located. Each octant corresponds to a unique combination of signs for the coordinates.

Question 64

The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in 3D space is calculated as $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Accepted Answer

Answer: True. Solution: This formula is derived from the Pythagorean theorem, considering the differences in each coordinate direction as the sides of right triangles.

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

Introduction to Three Dim..

Learning Objectives

Learning Objectives

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

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

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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Multiple Choice Questions

Q1.If the points P(−2,3,5)P(-2, 3, 5)P(−2,3,5), Q(1,2,3)Q(1, 2, 3)Q(1,2,3), and R(7,0,−1)R(7, 0, -1)R(7,0,−1) are given, what is the relationship between the distances PQPQPQ, QRQRQR, and PRPRPR?

Correct Answer: A

Q2.A point P(x,y,z)P(x, y, z)P(x,y,z) lies in the octant where x>0x > 0x>0, y<0y < 0y<0, and z>0z > 0z>0. Identify the octant in which the point lies.

Correct Answer: C

Q3.What is the equation of the set of points P(x,y,z)P(x, y, z)P(x,y,z) such that the sum of the distances from A(4,0,0)A(4, 0, 0)A(4,0,0) and B(−4,0,0)B(-4, 0, 0)B(−4,0,0) is equal to 10?

Correct Answer: A

Q4.Consider a point P(x,y,z)P(x, y, z)P(x,y,z) in the three-dimensional coordinate system. If PPP lies in the octant where x>0x > 0x>0, y<0y < 0y<0, and z<0z < 0z<0, which octant is PPP located in?

Correct Answer: C

Q5.Which of the following statements is true about the coordinates of a point on the YZ-plane?

Correct Answer: A

Q6.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

Q7.Which of the following conditions must be satisfied for three points P(x1,y1,z1)P(x_1, y_1, z_1)P(x1​,y1​,z1​), Q(x2,y2,z2)Q(x_2, y_2, z_2)Q(x2​,y2​,z2​), and R(x3,y3,z3)R(x_3, y_3, z_3)R(x3​,y3​,z3​) to be collinear in three-dimensional space?

Correct Answer: A

Q8.Given the point (4,−2,3)(4, -2, 3)(4,−2,3), identify the octant it lies in.

Correct Answer: B

Q9.If a point P(x,y,z)P(x, y, z)P(x,y,z) is equidistant from the x-axis, y-axis, and z-axis, which of the following must be true?

Correct Answer: A

Q10.Given a point P(x,y,z)P(x, y, z)P(x,y,z) lies in the YZ-plane, what can be inferred about its x-coordinate?

Correct Answer: B

Q11.Consider a point R(0,−5,7)R(0, -5, 7)R(0,−5,7) in three-dimensional space. Which plane does this point lie on?

Correct Answer: A

Q12.A point S(−2,−3,−4)S(-2, -3, -4)S(−2,−3,−4) is in the seventh octant. What is the sign of each coordinate in this octant?

Correct Answer: C

Q13.Find the coordinates of the fourth vertex DDD of a parallelogram ABCDABCDABCD given 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).

Correct Answer: A

Q14.Given a point P(x,y,z)P(x, y, z)P(x,y,z) in space, which of the following statements is true if PPP lies on the x-axis?

Correct Answer: A

Q15.If a point has coordinates (x,y,z)(x, y, z)(x,y,z) with x>0x > 0x>0, y<0y < 0y<0, and z<0z < 0z<0, which octant does it belong to?

Correct Answer: D

Q16.Calculate the distance between the points P(1,−3,4)P(1, -3, 4)P(1,−3,4) and Q(−4,1,2)Q(-4, 1, 2)Q(−4,1,2) in three-dimensional space.

Correct Answer: A

Q17.Given two points A(2,−3,4)A(2, -3, 4)A(2,−3,4) and B(−1,2,−5)B(-1, 2, -5)B(−1,2,−5) in a 3D space, what is the distance between them?

Correct Answer: B

Q18.Given two points P(1,−2,3)P(1, -2, 3)P(1,−2,3) and Q(4,2,−1)Q(4, 2, -1)Q(4,2,−1) in a 3D space, what is the distance between them?

Correct Answer: B

Q19.If points P(2,−1,3)P(2, -1, 3)P(2,−1,3), Q(5,2,6)Q(5, 2, 6)Q(5,2,6), and R(8,5,9)R(8, 5, 9)R(8,5,9) are given, find the condition for these points to be collinear.

Correct Answer: B

Q20.If a point has coordinates (x,y,z)(x, y, z)(x,y,z), what do these coordinates represent?

Correct Answer: B

Q21.In a three-dimensional coordinate system, if a point P(a,b,c)P(a, b, c)P(a,b,c) lies on the XZXZXZ-plane, what can be said about its yyy-coordinate?

Correct Answer: C

Q22.What is the significance of the origin in a three-dimensional coordinate system?

Correct Answer: A

Q23.Consider the points A(−1,0,1)A(-1, 0, 1)A(−1,0,1), B(2,3,4)B(2, 3, 4)B(2,3,4), and C(5,6,7)C(5, 6, 7)C(5,6,7). Which of the following statements is true regarding their collinearity?

Correct Answer: A

Q24.Which of the following points is equidistant from the origin and the point (4,0,0)(4, 0, 0)(4,0,0)?