2. Graphing in 3-D
Graphing in 3-D means that we need 3 coordinates to define a point
(x,y,z)
These are the coordinate planes, and they divide space into 8 octants. The “1st octant” is where all 3 coordinates are positive

3. It is difficult to define a point in space when graphing on a plane.
Ex. Graph the points (2,4,3), (2,-4,3), (-2,4,-3), and (2,4,0).

4. The projection of a point is like its shadow:
This system for graphing in 3-D is called rectangular, or Cartesian.

5. Thm. Distance Formula in Three Dimensions
The distance between (x1,y1,z1) and (x2,y2,z2) is
Ex. Find the distance between P(2,-1,7) and Q(1,-3,5).

6. Thm. Equation of a Sphere
The sphere with center (h,k,l) and radius r is

7. Ex. Find the equation of a sphere if a diameter has endpoints (-2,4,3) and (4,0,1).

8. Ex. Find the equation of a sphere with center (3,2,-5) that is tangent to the xy-plane.

9. Ex. Find the center and radius of the sphere x2 + y2 + z2 + 4x – 6y +2z + 6 = 0

10. Ex. What surfaces in are represented by the equations?
z = 3
y = 5

11. Ex. Sketch the surface in represented by the equation y = x.

12. Ex. Determine whether (3,-4,1), (5,-1,-1), and (1,-7,3) are collinear.

13. Ex. Find the distance from (3,-4,1) to
the xz-plane
the y-axis

14. Ex. What is the intersection of the sphere
and the yz-plane?

15. Ex. What region in is represented by the inequalities 1 ≤ x2 + y2 + z2 ≤ 4 and z ≥ 0?

16. Ex. Write inequalities to describe the region.
The half-space consisting of all points above the xy-plane.
The solid rectangular box in the first octant bounded by the planes x = 2, y = 9, and z = 4.