Download MAG OA1 (Algebra/Geometry) Course Summary

MAG OA1 (Algebra/Geometry) Course Summary
Unit #1: Vectors
- Vector Addition and Subtraction w/ Magnitudes and Angle
- triangle law & cosine law
- Scalar Multiplication w/ Magnitudes
- Vector Properties - commutative, associative, distributive, etc
- Addition and Subtraction w/ Components: (a, b, c)  (d , e, f )  (a  d , b  e, c  f )
- Scalar Multiplication w/ Components: k (a, b, c)  (ka, kb, kc)
- Finding Magnitudes with given Components: u  (a, b, c) | u |  a 2  b 2  c 2
- Dot Product w/ Magnitudes and Angle:
a  b | a | | b | cos
a  b  a x bx  a y b y  a z bz
- Dot Product w/ Components:
- Properties of the Dot Product - commutative, distributive
a b
- Scalar Projections:
scalar proj. =
|b|
a b
b
- Vector Projections:
vector proj. =
| b |2
Unit #2: Vector Applications I
- Systems of Equilibrium - resultant and equilibrant
- Velocity and Acceleration Problems
- Work:
W = F·s
- Linear Dependence for 2 Vectors: dependent = parallel; independent = not parallel
- Linear Dependence for 3 Vectors: dependent = coplanar; independent = not coplanar
- Theorem 1: u & v are independent, z = a u + b v where a, b  0
- Theorem 2: u, v, & w are dependent if they are coplanar
- Theorem 3: u, v, & w are independent, z = a u + b v + c w where a, b, c  0
- Division of a Line Segment: “C divides AB in ratio 5 : 2” means AC : CB = 5 : 2
n
m
OQ 
OR
- If P divs QR in ratio m : n, then OP 
mn
mn
Unit #3: Vector Applications II
- Given a picture, expressing one vector in terms of two others
- Finding the ratio dividing two lines within a picture - m : n & r : s
- Expressing the area of an interior shape as a fraction of the area of a larger shape
- Vector Proofs in Euclidean Geometry
- Cross Product w/ Components: a  b  (a2 b3  a3b2 , a3b1  a1b3 , a1b2  a2 b1 )
- Cross Product Magnitude w/ Magnitudes and Angle: | a  b |  | a | | b | sin 
- Area of a Parallelogram - the magnitude of the cross product of two sides
- Properties of Cross Product - anti-commutative, distributive, associative
Unit #4: Lines and Planes
- Lines in R2:
- Vector Equation:
- Parametric Equations:
- Cartesian Equation:
- Lines in R3:
- Vector Equation:
- Parametric Equations:
- Cartesian Equation:
- Symmetric Equations:
r  p  tm
x = ?, y = ? (from vector eqn)
ax  by  c  0 (where m = (-b, a) & n = (a, b) )
r  p  tm
x = ?, y = ?, z = ? (from vector eqn)
none
x  p1 y  p2 z  p3


m1
m2
m3
- Planes in R3:
r  p  sa  tb
- Vector Equation:
- Parametric Equations:
x = ?, y = ?, z = ? (from vector eqn)
ax  by  cz  d  0 (where n  (a, b, c) )
- Cartesian Equation:
- Finding the Angle Between a Line and a Plane
- Graphing Planes in R3
Unit #5: Intersection of Lines and Planes
- Intersection of 2 Lines in R3
- Intersection of 2 Planes
- Intersection of 3 Planes
- Intersection of 2 or 3 Planes using Matrices and Reduced Row Echelon Form
| ax1  by1  c |
d
- Distance between a Point and a Line in R2:
a2  b2
| PR  m |
- Distance between a Point and a Line in R3:
d
|m|
| ax1  by1  cz1  d |
d
- Distance between a Point and a Plane:
a2  b2  c2
P1 P2  (m1  m 2 )
d
- Distance between Skew Lines:
| m1  m 2 |
Unit #6: Linear Transformations and Matrices
- Linear Transformations - definition
( x, y )  ( x  y , y )
- Notation: ie:
T ( x, y )  ( x  y , y )
1  1  x   x  y 
T ( x)  
   

0 1   y   y 
- Drawing the Pictures of Linear Transformations with Unit Square
- Finding the Equation of a Line under a Transformation
- Adding, Scalar Multiplying, and Regular Multiplying of Matrices
- Composition of Transformations
Unit #7: Translations and Rotations
- Simple Translations: “along the vector (h, k)” means ( x, y )  ( x  h, y  k )
- sub (x - h) and (y - k) into original equation for image
- Basic Conics:
- Circle:
x2  y2  r 2
- Ellipse:
- Parabola:
- Hyperbola:
x2 y2

 1 or
a2 b2
- opening right:
- opening left:
- opening up:
- opening down:
x2 y2

1
b2 a2
y 2  ax
y 2  ax
x 2  ay
x 2  ay
x2 y2

 1 (crosses x-axis)
a2 b2
or
where a > b
x2 y2

 1 (crosses y-axis)
b2 a2
- Translating Conics:
sub (x - h) and (y - k) in
- Undoing a Translation:
complete the square - if there is no xy terms!
- Rotations - matrix form:
orginal: XtAX = c, image: UtBU = c where B = RtAR
eqn: ax 2  2h xy  by 2  c
a h 
A

h b 
cos
R  
 sin 
 sin  
cos 
 cos
R  R  
 sin 
sin  
cos 
ellipse/circle: ab - h2 > 0
parabola:
ab - h2 = 0
hyperbola:
ab - h2 < 0
2
tan 2 
- Elimination of the xy Term - to find :
but if a = b,  = 45o
ba
- If q isn’t a nice number.... use tan 2 to find cos 2, then use:
1  cos 2
1  cos 2
sin 2  
cos 2  
2
2
- Sketching Complex Quadratic Equations - rotate then translate to get the basic form
- Determining Conic Type:
Unit #8: Mathematical Induction
- Mathematical Induction Procedure:
1) Prove P1 is true
2) Assume Pk is true and prove Pk+1 is true
- Induction using Sigmas
n
- Properties of Sigma:
 k  kn
j 1
n
n
j 1
j 1
k a j  ka j
n
 (a
j 1
j
n
n
j 1
j 1
 bj )   a j  bj
n!
(n  r )!
n
n!
C (n, r )    
 r  (n  r )! r!
P(n, r ) 
- Permutations: when order matters:
- Combinations: when order doesn’t matter:
- Pascal’s Triangle
 n   n  1  n  1
   
  

 r   r  1  r 
n
n
- The Binomial Theorem:
(a  b) n    a n r b r
r 0  r 
n(n  1) 2 n(n  1)( n  2) 3
x 
x  ...
- Binomials w/ Real Exponents: (1  x) n  1  nx 
2!
3!
- Pascal’s Law:
Unit #9: Complex Numbers
- Cartesian Complex Number Form:
- Real Part: Re (a + bi) = a
a + bi
where a, b  R and i   1
- Imaginary Part: Im (a + bi) = b
- Cartesian Graphing of Complex Numbers
- Addition and Subtraction (Cartesian): ie: (3 - 7i) + (8 + 2i) = 11 - 5i
- Multiplication (Cartesian): ie: (3 - 7i) (8 + 2i) = 24 + 6i - 56i + 14 = 38 - 50i
- Division (Cartesian): multiply top and bottom by the bottom’s complex conjugate
- Modulus (Cartesian):
| a + bi | =
a2 + b2
- Equality of 2 Complex Numbers (Cartesian) - solving basic equations
- Solving Complex Equations for Complex Roots:
- fundamental theorem of algebra - “an equation of degree n has n roots”
x2 - (sum of roots) x + (product of roots) = 0
r cis  for r the radius and  the rotation angle
y  r sin 
x  r cos
y
- Cartesian ® Polar Conversions:
r 2  x 2  y 2 tan  
x
- Polar Graphing of Complex Numbers
- Polar Complex Number Form:
- Polar ® Cartesian Conversions:
- Multiplication (Polar):
- Division (Polar):
(r1cis1 )(r2 cis 2 )  r1r2 cis (1   2 )
(r1cis 1 ) r1
 cis ( 1   2 )
(r2 cis 2 ) r2
- Equality of 2 Complex Numbers (Polar)
- Exponential Complex Number Form:
rcis   r e i
- De Moivre’s Theorem:
(r cis ) n  r n cis (n )
- Roots of Complex Numbers - cartesian and polar ways
R.W. Newson, 2000.
ie: e i  1
nZ