Three Dimensional Geometry and Vectors Dr. Shildneck

2-D vs 3-D  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).

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.

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

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

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

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

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

MAGNITUDE Find the magnitude of each vector i – 9j + 4k

Operations with 3-D Vectors Given x = and y = Find 3x – 2y

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