Introduction to Three Dimensional Geometry

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.

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	+	-	-

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.