1. Three Dimensional Geometry and Vectors
Dr. Shildneck

2. Two Dimensional Geometry is done in/on a flat surface called a PLANE.
Three Dimensional Geometry is done in SPACE.
Two Dimensions require two axes, x and y.
Three Dimensions require three axes, x, y and z.
Two Dimensional locations are indicated by a coordinate pair (x, y).
Three Dimensional locations are denoted by a coordinate triple (x, y, z).
2-D vs 3-D

3. The 3-Dimensional Coordinate System
Traditionally, when representing
The 3-D Cartesian Coordinate System,
The x-axis is back to front,
The y-axis is left to right, and
The z-axis is down to up.

4. Plotting Points Plotting 3-D points Just like 2-D, move the
proper amounts (from the origin),
in alphabetical order, to get to the
appropriate location.
Example
Plot: (3, 2, 1)
Plotting Points

5. Plotting Points Plotting 3-D points Another way of doing it
Is to draw a box…
Example
Plot: (-2, 3, 2)
Plotting Points

6. 3-D Distance Find the distance from (-2,-3, 0) to (3, 3, 3)
3-D Distance Formula

7. Find point half way from (-2,-3, 0) to (3, 3, 3) – AKA (The Midpoint)
3-D Midpoint
Midpoint Formula
Average the x’s, y’s, and the z’s

8. Plotting 3-D Vectors Plotting 3-D Vectors (in standard position)
Use the box method…
Example
Draw <-2, 3, 2>
Plot the point
Draw vector (in
standard position)
from (0, 0, 0) to
the point.
Plotting 3-D Vectors

9. Writing 3-D Vectors Writing 3-D Vectors in
(in component form and as a linear combination)
Examples
Write the vector with initial point A and terminal point B.
1) A(2, -5, 6), B(8, 11, 12) in component form
2) A(-3, 2, 2), B(0, 5, 6) as a linear combination
Writing 3-D Vectors

10. MAGNITUDE Find the magnitude of each vector. <4, 7, -4>
3i – 9j + 4k
MAGNITUDE

11. Operations with 3-D Vectors
Given x = <4, 7, -4> and y = <2, 3, 7>
Find 3x – 2y
Operations with 3-D Vectors

12. ASSIGNMENT Assignment YOUR (yellow) TEXTBOOK Section 8-4 Page. 514-515
#1-13 (odd), (odd), (odd), 51, 53
ASSIGNMENT