1. Chapter 12
Math 181

2. An equation of a sphere is

3. Vectors

4. Vectors

5. Cross Product

6. Lines
A line in 3 dimensional space is determined when we know a point P0(x0,y0,z0) on L and the direction of L.
In three dimensions, the direction of a line is conveniently described by a vector, so we let v be a vector parallel o L.
Let P(x, y, z) be an arbitrary point on L and let r0 and r be the position vectors of P0 and P.

8. But since a and v are parallel vectors, there is a scalar t such that
If a is the vector with representation P0P, then the Triangle Law for vector addition gives
But since a and v are parallel vectors, there is a scalar t such that
Vector Equation of L

9. As t varies, the line is traced out by the tip of the vector

10. Parametric Equations of L

11. where a, b, c are the direction numbers of L
Symmetric Equations of L

12. Planes
A plane in space is determined by a point P0(x0,y0,z0) in the plane and a vector n that is orthogonal to the plane.
The orthogonal vector n is called a normal vector.
Let P(x, y, z) be an arbitrary point in the plane and let r0 and r be the position vectors of P0 and P.

13. Then the vector r0 - r is represented by P0P
Then the vector r0 - r is represented by P0P. The normal vector n is orthogonal to every vector in the given plane.
In particular, n is orthogonal to r0- r.

14. Scalar Equation of the plane
Linear Equation of the plane

15. Two planes are parallel if their normal vectors are parallel.
If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their normal vectors.

16. Example
Find an equation of the plane through the points (3, -1, 2), (8, 2, 4), and (-1, -2, -3).

17. Example
Find an equation of a plane that passes through the point (-1, 2, 1) and contains the line of intersection of the planes

18. Cylinders
A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve.
When you are dealing with surfaces, it is important to recognize that an equation like represent a cylinder not a circle.

19. The trace of the cylinder
in the xy-plane is the circle

20. Quadric Surfaces
A quadric surface is the graph of a second degree equation in the three variables x, y, z

21. Ellipsoid All traces are ellipses.
If a = b = c, the ellipsoid is a sphere.

22. Section 13.6 · Table 1a All traces are ellipses.
If a = b = c, the ellipsoid is a sphere.
Section · Table 1a A

23. Elliptic Paraboloid Horizontal traces are ellipses.
Vertical traces are parabolas.
Variable raised to the 1st power indicates the axis of the paraboloid.

24. Elliptic Paraboloid

25. Hyperbolic Paraboloid
Horizontal traces are hyperbolas.
Vertical traces are parabolas.
The shape resembles a saddle.

26. Hyperbolic Paraboloid

27. Cone
The horizontal traces are ellipses. The vertical traces in the place x = k and y = k are hyperbolas if k  0 but are pairs of lines if k = 0.
The cone opens about the axis of the variable by itself or the variable with the negative sign.

28. Cone

29. Hyperboloid of One Sheet
Horizontal traces are ellipses.
The vertical traces are hyperbolas.
The axis of symmetry corresponds to the variable whose coefficient is negative.

30. Hyperboloid of One Sheet

31. Hyperboloid of Two Sheets
The axis of symmetry corresponds to the variable whose coefficient is positive.

32. Hyperboloid of Two Sheets

33. Cylindrical Coordinate System
P(r, , z)
(r, , 0)  x y z

34. Cylindrical Coordinate System
In the cylindrical coordinate system, a point in 3 dimensional space is represented by the ordered triple (r, , z) where r and  are the polar coordinates of the projection of P on the xy-plane and z is the directed distance from the xy-plane to P.

35. Cylindrical Coordinate System

36. Spherical Coordinate System x y z  
= c sphere
= c half cone
 = c half plane

37. Spherical Coordinate System
The spherical coordinates (, , ) of a point P in space where  = |OP| is the distance from the origin to P,  is the same angle as in cylindrical coordinates, and  is the angle between the positive z-axis and the line segment OP.

38. Spherical Coordinate System