Three-Dimensional Coordinate Systems, Distance Formulas, and Spheres

Dimensional Spaces and Coordinate Systems

  • One-Dimensional Space (R1\mathbb{R}^1):

    • Represented using a single real-number line with a single variable, typically xx.

    • Set notation designation: x∈Rx \in \mathbb{R}.

    • Graphing in 1D space involves placing tick marks along the xx-axis and marking or shading relevant intervals (e.g., for the inequality x>3x > 3, placing a tick mark at 33 and shading all points strictly greater than 33 along the axis).

  • Two-Dimensional Space (R2\mathbb{R}^2):

    • Represented on the xyxy-plane, formed by the intersection of a horizontal xx-axis and a vertical yy-axis.

    • The origin in 2D space is defined by the ordered pair (0,0)(0, 0).

    • Points in this plane are defined as ordered pairs (x,y)(x, y).

    • Set notation designation: {(x,y)∣x∈R,y∈R}\{(x, y) \mid x \in \mathbb{R}, y \in \mathbb{R}\}, which combines two sets of real numbers and is written as R2\mathbb{R}^2.

  • Three-Dimensional Space (R3\mathbb{R}^3):

    • Combines three mutually perpendicular coordinate axes: the xx-axis, yy-axis, and zz-axis.

    • The origin in 3D space is defined by the ordered triple (0,0,0)(0, 0, 0), representing the common intersection point of all three axes.

    • Points in 3D space are defined as ordered triples (x,y,z)(x, y, z).

    • Set notation designation: {(x,y,z)∣x∈R,y∈R,z∈R}\{(x, y, z) \mid x \in \mathbb{R}, y \in \mathbb{R}, z \in \mathbb{R}\}, representing three sets of real numbers combined as R3\mathbb{R}^3.

    • Hand Visualization Rule for Right-Hand Axis Orientation:

      • Extend the pointer finger straight ahead to represent the positive xx-axis.

      • Curl the remaining three fingers inward at a right angle to represent the positive yy-axis.

      • Extend the thumb perpendicular to both the pointer finger and the curled fingers to represent the positive zz-axis.

Coordinate Planes and Space Division

  • The Three Primary Coordinate Planes:

    • xyxy\text{-plane} (z=0z = 0): Formed by the intersection of the xx-axis and yy-axis; contains all points where the zz\text{-coordinate} is zero.

    • yzyz\text{-plane} (x=0x = 0): Formed by the intersection of the yy-axis and zz-axis; contains all points where the xx\text{-coordinate} is zero.

    • xzxz\text{-plane} (y=0y = 0): Formed by the intersection of the xx-axis and zz-axis; contains all points where the yy\text{-coordinate} is zero.

  • Planes Parallel to Coordinate Planes:

    • x=kx = k: Represents a plane parallel to the yzyz\text{-plane} (x=0x = 0). Sliding this plane along the xx\text{-axis} corresponds to varying the constant value kk.

    • y=ky = k: Represents a plane parallel to the xzxz\text{-plane} (y=0y = 0), slidable along the yy\text{-axis}.

    • z=kz = k: Represents a plane parallel to the xyxy\text{-plane} (z=0z = 0), slidable along the zz\text{-axis}.

  • Space Division by Infinite Planes:

    • 11 Infinite Plane: Divides three-dimensional space into 22 regions.

    • 22 Infinite Planes:

      • Non-intersecting (Parallel) Planes: Divide space into 33 regions.

      • Intersecting Planes: Divide space into 44 regions.

    • 33 Infinite Planes:

      • Three Parallel Planes: Divide space into 44 regions.

      • Two Parallel Planes Intersected by a Third Plane: Divide space into 66 regions.

      • Three Planes Intersecting Pairwise (No Single Shared Line or Point): Divide space into 77 regions.

      • Three Mutually Intersecting Planes at a Single Point: Divide space into 88 regions.

Octants of Three-Dimensional Space

  • Definition and Naming Conventions:

    • The 88 distinct regions formed by the mutual intersection of the three coordinate planes (xyxy, yzyz, xzxz) are called octants.

    • Unlike the four quadrants of 2D space (Quadrants I, II, III, IV), the octants in 3D space are not all assigned standard numerical names because remembering a specific sequence for eight 3D regions is difficult.

    • The only octant given a standardized numerical name is the First Octant, defined as the region where all three coordinates are strictly positive: (+,+,+)(+, +, +).

    • All other octants are uniquely identified by the combination of signs of their (x,y,z)(x, y, z) coordinates.

  • The Eight Coordinate Sign Combinations:

    • (+,+,+)(+, +, +) — First Octant

    • (+,+,−)(+, +, -) — Represents the bottom front right region (where moving right yields positive xx, moving forward yields positive yy, and moving downward yields negative zz)

    • (+,−,+)(+, -, +)

    • (+,−,−)(+, -, -)

    • (−,+,+)(-, +, +)

    • (−,+,−)(-, +, -)

    • (−,−,+)(-, -, +)

    • (−,−,−)(-, -, -)

Three-Dimensional Distance and Midpoint Formulas

  • Three-Dimensional Distance Formula:

    • Derived from the Pythagorean theorem extended from two dimensions into three dimensions.

    • In 2D space, the distance dd between (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:         d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

    • In 3D space, adding the distance along the zz\text{-axis} yields the distance dd between points (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2):         d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

  • Three-Dimensional Midpoint Formula:

    • Calculates the center point (xm,ym,zm)(x_m, y_m, z_m) located exactly midway between two endpoints (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2) by averaging each coordinate component:         (xm,ym,zm)=(x1+x22,y1+y22,z1+z22)(x_m, y_m, z_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)

Equations and Geometry of Spheres

  • Definition of a Sphere:

    • A sphere is the set of all points in three-dimensional space that are at a constant distance rr (radius) away from a fixed central point designated as (h,k,l)(h, k, l).

    • It is the three-dimensional equivalent of a 2D circle on a flat plane.

  • Standard Equation of a Sphere:

    • Obtained by squaring both sides of the 3D distance formula applied between a variable surface point (x,y,z)(x, y, z) and the fixed center (h,k,l)(h, k, l):         (x−h)2+(y−k)2+(z−l)2=r2(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

  • Tangency to Coordinate Planes:

    • A sphere is tangent to a coordinate plane if it touches that plane at exactly one point.

    • If a sphere is tangent to the yzyz\text{-plane} (x=0x = 0), the radius rr is equal to the absolute value of the xx\text{-coordinate} of the sphere's center:         r=∣h∣r = |h|

Worked Examples and Algebraic Techniques

  • Example 1: Finding the Equation of a Sphere Given Center and Tangency:

    • Problem: Write the standard equation of a sphere with center (h,k,l)=(−2,4,−6)(h, k, l) = (-2, 4, -6) that is tangent to the yzyz\text{-plane}.

    • Step 1: Identify center parameters: h=−2h = -2, k=4k = 4, l=−6l = -6.

    • Step 2: Calculate the radius. Since the sphere is tangent to the yzyz\text{-plane} (x=0x = 0), the distance from the center's xx\text{-value} (−2-2) to x=0x = 0 dictates the radius:         r=∣−2∣=2r = |-2| = 2         r2=22=4r^2 = 2^2 = 4

    • Step 3: Substitute hh, kk, ll, and r2r^2 into the standard equation:         (x−(−2))2+(y−4)2+(z−(−6))2=4(x - (-2))^2 + (y - 4)^2 + (z - (-6))^2 = 4         (x+2)2+(y−4)2+(z+6)2=4(x + 2)^2 + (y - 4)^2 + (z + 6)^2 = 4

  • Example 2: Finding the Equation of a Sphere Given Diameter Endpoints:

    • Problem: Find the standard equation of a sphere whose diameter has endpoints at (−1,2,3)(-1, 2, 3) and (5,−2,7)(5, -2, 7).

    • Step 1: Compute the center (h,k,l)(h, k, l) using the 3D midpoint formula between the endpoints:         h=−1+52=42=2h = \frac{-1 + 5}{2} = \frac{4}{2} = 2         k=2+(−2)2=02=0k = \frac{2 + (-2)}{2} = \frac{0}{2} = 0         l=3+72=102=5l = \frac{3 + 7}{2} = \frac{10}{2} = 5         Center (h,k,l)=(2,0,5)\text{Center } (h, k, l) = (2, 0, 5)

    • Step 2: Compute the radius rr as the distance between the center (2,0,5)(2, 0, 5) and one endpoint (−1,2,3)(-1, 2, 3):         r=(−1−2)2+(2−0)2+(3−5)2r = \sqrt{(-1 - 2)^2 + (2 - 0)^2 + (3 - 5)^2}         r=(−3)2+(2)2+(−2)2r = \sqrt{(-3)^2 + (2)^2 + (-2)^2}         r=9+4+4=17r = \sqrt{9 + 4 + 4} = \sqrt{17}         r2=17r^2 = 17

    • Step 3: Write the standard equation:         (x−2)2+(y−0)2+(z−5)2=17(x - 2)^2 + (y - 0)^2 + (z - 5)^2 = 17

  • Example 3: Converting Expanded Form to Standard Form via Completing the Square:

    • Problem: Given the expanded sphere equation x2+4x+y2−2y+z2+2z−2=0x^2 + 4x + y^2 - 2y + z^2 + 2z - 2 = 0, find its center and radius.

    • Step 1: Rearrange terms by variable groups and move the constant term to the right side:         (x2+4x)+(y2−2y)+(z2+2z)=2(x^2 + 4x) + (y^2 - 2y) + (z^2 + 2z) = 2

    • Step 2: Complete the square for each variable grouping:

      • For xx: add (42)2=4\left(\frac{4}{2}\right)^2 = 4

      • For yy: add (−22)2=1\left(\frac{-2}{2}\right)^2 = 1

      • For zz: add (22)2=1\left(\frac{2}{2}\right)^2 = 1

    • Step 3: Add the same squared values to the right side to maintain equality:         (x2+4x+4)+(y2−2y+1)+(z2+2z+1)=2+4+1+1(x^2 + 4x + 4) + (y^2 - 2y + 1) + (z^2 + 2z + 1) = 2 + 4 + 1 + 1

    • Step 4: Factor into binomial squares and simplify the right side sum:         (x+2)2+(y−1)2+(z+1)2=4(x + 2)^2 + (y - 1)^2 + (z + 1)^2 = 4

    • Step 5: Extract the center (h,k,l)(h, k, l) and radius rr:

      • Center (h,k,l)=(−2,1,−1)\text{Center } (h, k, l) = (-2, 1, -1)

      • Radius r=4=2\text{Radius } r = \sqrt{4} = 2