Math 223
Outline of material for Test 1
Chapter 1. Vectors and the Geometry of Space
§12.1: Three-Dimensional Coordinate Systems
3D coordinates, coordinate axes, and coordinate planes
equations of a plane parallel to one of the coordinate planes
Distance formula
Equation of a sphere
§12.2: Vectors
Definition
Displacement vector
Initial point (tail) and terminal point (head or tip)
Zero vector
Vector addition – Triangle and Parallelogram Laws
Scalar multiplication
Negative of a vector
Vector subtraction
Vector components
Components of a displacement vector
Position vector
Magnitude (Length) of a vector
Properties of vectors (Table p. 774)
Standard basis vectors î, ĵ, k
unit vectors
Applications: Sum of force and velocity vectors
§12.3: The Dot Product
Definition (algebraic)
Properties (2)
Geometric interpretation (3)
Angle in terms of dot product (6)
Orthogonality condition (7)
Direction angles & cosines (8,9,10,11)
Scalar projection of b onto a (scalar = signed magnitude of b in the direction of a):
compa(b) = b · unit(a)
Vector projection of b onto a (vector = the above scalar times the vector unit(a) ):
proja(b) = compa(b) unit(a) = ( b · unit(a) ) unit(a)
where unit(a) = a / ||a|| , the unit vector in the direction of a.
Application: Work
§12.4 The Cross Product
Definition (algebraic)
Determinants
Direction & Magnitude of cross product (5,6)
Right-hand rule
Parallelism condition (7)
Magnitude = area of parallelogram
Properties (8)
Scalar Triple Product
Volume of Parallelepiped
Application: Torque
Math 223
Outline of material for Test 1
§12.5: Equations of Lines and Planes
Line is determined by point and direction vector
Equations of lines:
- Vector (1)
- Parametric (2)
- Symmetric (non-parametric) (3)
Equation of line through two points (Example 2)
Line segment (4)
Parallel lines
Looking for intersection of non-parallel lines –
system of 3 equations in the 2 line parameters
Skew lines: non-parallel lines that do not intersect
Plane is determined by point and normal vector
Equations of lines:
- Vector (5,6)
- Scalar (7,8)
Equation of plane through three points (Example 5)
Intersection point of line & plane (Example 6)
Parallel planes
Angle between planes
Line of intersection of pair of planes (Example 7b)
Distance of a point to a plane (Example 8; uses scalar projection)
Distance between parallel planes (Example 9; builds on previous example)
Distance between skew lines (Example 10; builds on previous example)
§12.6: Cylinders and Quadric Surfaces
Traces
Definition of cylinder
(Confine attention to case of rulings parallel to a coord. axis: algebraically, equation omits 1 of x,y,z)
Quadric Surfaces
General equation (bottom p. 805)
All traces of Quadric Surfaces are Conic Sections (includes degenerate ones)
Two main cases
* Ellipsoids & Hyperboloids: A,B,C,J ≠ 0 (Coefficients refer to those in eqn. on bottom p. 805)
- Ellipsoids: A,B,C same sign, J other sign – all traces are ellipses (generalizes Sphere)
- Hyperboloids: A,B,C not all same sign
Sign of A,B,C that disagrees with the other two gives direction of axis
Traces normal to axis are ellipses, others are hyperbolas
- 1 sheet: exactly 1 of signs of A,B,C agrees with that of J
- 2 sheets: exactly 2 of signs of A,B,C agree with that of J
- Cones are degenerate Hyperboloids with J=0.
* Paraboloids: A,B,I or B,C,G or A,C,H ≠ 0
Missing one of A,B,C gives direction of axis;
Traces containing the axis direction are parabolas; ones normal depend on type of Paraboloid:
- Elliptic: A,B (or B,C or A,C) same sign; traces normal to axis are ellipses
- Hyperbolic: A,B etc. opposite sign; traces normal to axis are hyperbolas
Any of these can be translated by replacing x by x–a, etc., giving rise to additional x,y,z terms.
A given equation may be recognized as translated standard form by completing the square