LHE 11.2 Three-Dimensional Coordinate in Space Calculus III Berkley High School September 14, 2009

2 Conventions of Three-Dimensional Space With x, y and z axes perpendicular to each other in three dimensional space, each (a,b,c) of real numbers corresponds to a unique point in space. Right-Hand Rule…

3 Examples Graph points A:(0,5,0), B:(5,4,6), C:(1,-1,3)

4 Equations in R 3 What does z=1 look like? {(x,y,1)|x,y are R} A plane of height 1 above the xy plane

5 Equations in R 3 What does y=1 look like? {(x,1,z)|x,z are R} A plane of distance 1 unit right the xz plane

6 Equations in R 3 What does x=2 look like? {(2,y,z)|y,z are R} A plane parallel to the yz plane and two units in the positive x direction

7 Equations in R 3 What does x=y look like? {(x,x,z)|x,z are R} A vertical plane that crosses through the xy plane through the line x=y

8 Equations in R 3 xyz=0 {(x,y,z)|x=0 or y=0 or z=0} yz plane union xz plane union xy plane

9 Distance between a point and the origin Find the distance between the origin the point (1,2,3). Find the distance between the origin and the point (x, y, z).

10 Distance formula from origin to any point in R 3

11 Distance between any two points in R 3 Find the distance between the given one point A: (x 1, y 1, z 1 ) and point B: (x 2, y 2, z 2 ). If we translate A to the origin then adjust B accordingly, we can use the earlier formula.

12 Defining a sphere Definition of a sphere centered at the origin: all points equidistant from particular point (center).

13 Defining a sphere Definition of a sphere centered at the (a,b,c): all points equidistant from particular point (center).

14 What shape is this? A sphere centered at (2,-1,0) with radius 5^.5

15 Midpoint formula for R 3

16 Vectors in Component Notation

17 Directed Line Segment vs. Vector

18 Special Unit Vectors

19 Are two vectors parallel?

20 Are three points collinear?

21 Assignment Section 11.2, 1-67, odd, x61.