8-1 Polar Coordinates
I. Polar Coordinates
A.
B.
p. 590: 9-19 odd, 29-55 odd
B. The Polar Plane
II. Converting Polar to Rectangular
Given (𝑟, 𝜃) , then
x  r cos and y  r sin 
III. Converting Rectangular to Polar
Given P(𝑥, 𝑦) , then
y
y
r  x 2  y 2 and   tan 1 or tan 1   if x  0
x
x
IV. Renaming Polar Coordinates
(𝑟, 𝜃) is equivalent to:
A) (𝑟, 𝜃 ± 2𝜋𝑛)
B) (−𝑟, 𝜃 ± 𝜋)
Plot each point given in polar coordinates.
1. (6, 50°)
2. (−2,120°)
𝜋
3. (3, 4 )
Convert the polar coordinates to rectangular coordinates.
5𝜋
5. (2, 6 )
6. (1,3)
7. (2,40°)
Convert the rectangular coordinates to polar coordinates
8. (1, −1)
9. (−1, √3)
10. (−2.1, −1.5)
4. (5, −40°)
8-2 Polar Equations
Worksheet 1-15 all.
I. Polar Equations and Graphs
A. Lines
1.   
2. r cos  a
3. r sin  b
B. Circle
1. r  a
2. r  a cos
3. r  a sin
C. Cardioid
1. r  a  a cos
2. r  a  a sin
D. Limocon
1. a  b cos
2. a  b sin
E. Lemniscate
1. 𝑟 2 = 𝑎2 𝑐𝑜𝑠(2𝜃)
2. 𝑟 2 = 𝑎2 sin⁡(2𝜃)
F. Spiral of Archimedes
r  a , must be in radians
G. Rose
A. Odd Angle
1. 𝑟 = asin⁡(3𝜃)
2. 𝑟 = acos⁡(3𝜃)
Examples on a handout
B. Even Angle
1. 𝑟 = asin⁡(2𝜃)
2. 𝑟 = acos⁡(2𝜃)
Polar Graphs and Systems of Polar Equations
Polar Graphs and Systems Worksheet
I. Lines in Polar Form
A.   
B. r cos   a
C. r sin   b
II. Circles in Polar Form
A. r  a
B. r  a cos 
C. r  a sin 
III. Cardiod
A. r  a  a cos 
IV. Rose
A. r  a cos2 
B.
r  a sin 2 
Finding Rose Petal Tips
1. For odd angles, replace r
with a and solve for 
2. For even angles, replace
r with  a and solve for 
B. r  a  a sin 
r  a cos3 
r  a sin 3 
Finding Rose Petal Widths
Replace r with 0 and solve
for 
Convert each equation to polar form.
1. 5x 2  5y 2  7
2. y 2  3x
3. y  6
Convert each equation to rectangular form
4. r  sin
5. r 2  sin 
6. r  5
Graph each polar equation.
9. r  2cos
10. r  5  5sin
12. r  3
13. r  4 sin 3
7. r cos  8
8.  

6
11. r  5 sec 
Find the intersection of the two polar curves. Express the solution in the form r,   .
14. r  1 cos  and r  1 cos 
15. r  2  3 cos  and r  cos 
16. r  4 sin  and r  2
8-3 The Complex Plane and De Moivre’s Theorem
I.
p. 617: 1-5 odd, 11-15 odd, 21, 23, 27, 31, 33, 39
Polar Form of a Complex Number
Rectangular: z  x  yi
Polar
: z  r cos  r sini
II.
Product and Quotient of Complex Numbers
If 𝑧1 = 𝑟1 (𝑐𝑜𝑠𝜃1 + 𝑖𝑠𝑖𝑛𝜃1 ) and 𝑧2 = 𝑟2 (𝑐𝑜𝑠𝜃2 + 𝑖𝑠𝑖𝑛𝜃2 ) , then:
A. 𝑧1 𝑧2 = 𝑟1 𝑟2 [𝑐𝑜𝑠(𝜃1 + 𝜃2 ) + 𝑖𝑠𝑖𝑛(𝜃1 + 𝜃2 )]
𝑧
𝑟
B. 1 = 1 [𝑐𝑜𝑠(𝜃1 − 𝜃2 ) + 𝑖𝑠𝑖𝑛(𝜃1 − 𝜃2 )]
𝑧2
III.
𝑟2
De Moivre’s Theorem
If 𝑧 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃), then 𝑧 𝑛 = 𝑟 𝑛 [𝑐𝑜𝑠(𝑛𝜃) + 𝑖𝑠𝑖𝑛(𝑛𝜃)]
Write each complex number in polar form. Express the argument in degrees.
1. 3  i
2. 2i
3. –5
4. 1  3i
Write each complex number in rectangular form.
𝜋
𝜋
5. 5(𝑐𝑜𝑠150° + 𝑖𝑠𝑖𝑛150°)
6. 3 (𝑐𝑜𝑠 2 + 𝑖𝑠𝑖𝑛 2 )
z
. Leave answers in polar form.
w
𝑧 = 3(𝑐𝑜𝑠60° + 𝑖𝑠𝑖𝑛60°)
𝑤 = 4(𝑐𝑜𝑠30° + 𝑖𝑠𝑖𝑛30°)
7. Find zw and
Write the expression in standard x  yi form.
8. [3(𝑐𝑜𝑠30° + 𝑖𝑠𝑖𝑛30°)]
1
9. (2 +
3
√3
𝑖)
2
8-4 Vectors
p. 628: 1-15 odd, 19, 25, 27, 31-41 odd, 49, 51
I. Vector – a quantity that has both magnitude and direction.

A. Magnitude is length of the vector.
v
B. Direction is the angle formed with the x-axis and notated with an arrow.
II. Geometric Vectors
A. Directed line segment - PQ
1. Referred to as a geometric vector
2. P is the initial point and Q is the terminal point
B. Zero Vector – a vector with a magnitude of 0.
C. Equal vectors – vectors of the same magnitude and the same direction.
III. Algebraic Vectors


A. v  a, b where a,b are scalars called the components of v

B. Position vector – a vector v that has its initial point at the origin of a coordinate plane and
the terminal point is at a, b .

C. If v has initial point P1  x1 , y1  and terminal point P2  x2 , y 2 



And v  P1 P2 , then v is equal to the position v ector v  x 2  x1 , y 2  y1
IV. Alternate Form of Algebraic Vector

v  a, b  a 1,0  b 0,1  ai  bj
A. a is the horizontal component
B. b is the vertical component
V. Properties of Algebraic Vectors


Let v  a1i  b1 j  a1 , b1 and w  a 2 i  b2 j  a 2 , b2 and  be a scalar, then:


A. v  w  a1  a 2 i   y1  y 2  j  a1  a 2 , b1  b2

B.  v  a1 i  ab1  j  a1 , b1


C. The magnitude of v is v  a1  b1
2

2

E. a unit vector in the same direction of v is u 

v

v
VI. Vector in Terms of Magnitude and Direction


A. Given magnitude v of a nonzero vector v and angle  , 0    360 , then

v  v cos i  sin j 
B. Velocity Vector - represents speed and direction
Use the given vectors to graph the following vectors.



1. v  w


3. v  w
2. 2 v
Determine whether each statement is true.




4. A B  E


5. A C  0




6. D A B  C


7. The vector v has initial point P and terminal point Q. Write v in ai  bj form, that is, find its position
vector.
𝑃(−3,1) and 𝑄(2, −3)

8. Find v

v  3i  4 j


Find each quantity if v  2i  8 j and w  4i  6 j


9. 3 v  2 w


10. v  w


12. Find the unit vector having the same direction as v

v 2j

11. v  w

13. Write vector v in ai  bj form.

v  3 and   30
8-5 The Dot Product
p. 636: 1-11 odd
I. The Dot Product


 
If v  a1i  b1 j and w  a2i  b2 j , then v  w 
II. Properties of the Dot Product



If u, v , and w are vectors, then
 
A. u v 
B. 𝑢
⃗ ∙ (𝑣 + 𝑤
⃗⃗ ) =
 
C. v  v 
 
D. 0 v 
 
E. If v  w  0, then


F. If v 1  k w, then
III. Angle Between Vectors




If u and v are nonzero vectors, then the angle  between u and v where 0     is ________________
A. If   0 or  , then

B. If   , then
2


Find each of the following of u and v .
 
a) u v

 
b) the angle between u v

1. u  3i  j and v  6i  2 j


3. Find a so that v and w are orthogonal.


v  6i  10 j and w  ai  5 j


c) whether u and v are are parallel, orthogonal or neither


2. u  2i  j and v  3i  6 j
8-6 Vectors in Space
p. 647: 9, 13, 21, 25, 27, 29, 33, 35, 37, 39, 41, 45, 47
I. Distance Formula in Space
If 𝑃1 = (𝑥1, 𝑦1 , 𝑧1 ) and 𝑃2 = (𝑥2 , 𝑦2 , 𝑧2 ) , then the distance between them is:
d=
II. Vector in Space


If 𝑃1 = (𝑥1, 𝑦1 , 𝑧1 ) and 𝑃2 = (𝑥2 , 𝑦2 , 𝑧2 ) , then the position vector v  P1 P2  ai  bj  ck where

a  x2  x1 , b  y2  y1 , c  z2  z1 and a, b, c are the components of v .
III. Properties of Vectors


Let v  a1 , b1 , c1 and w  a2 , b2 , c2 and  is a scalar, then:




A. v  w 
B. v  w 

C.  v 

D. v 
 
E. v  w 

F. Unit Vector of v is

v

v

1. Find the Distance from P1 to P2 .
𝑃1 = (3,0, −4) and 𝑃2 = (5,4,2)

3. Find v

when v  2i  3 j  k
2. Find the position vector of v with initial point P
and terminal point Q.
𝑃 = (−4,3,1) and 𝑄 = (5,6, −2)


4. Find each value if v  3i  5 j  2k and w  i  2 j  3k .




b. v  w
a. 3 v  2 w

5. Find the unit vector having the same direction as v .

v  4i  j  4k


6. Find the dot product and the angle between v and w .


v  3i  8 j  k and w  4i  j  4k
8-7 The Cross Product
Pg. 653: 1, 5, 17-35 odd, 43
I. Two x Two Determinant
a1 b1
= a1b2  a2b1
a2 b2
II. Three x Three Determinant
A B C
b c
a
a1 b1 c1  A 1 1  B 1
b2 c2
a2
a2 b2 c2
c1
c2
C
a1
b1
a2 b2
III. The Cross Product
i
j k


 
If v  a1i  b1 j  c1k and w  a2i  b2 j  c2 k , then v  w  a1 b1 c1
a2 b2 c2
IV. Properties of the Cross Product
  
If u , v , w are vectors in space and  is a scalar, then
  
A. u  u  0
 
 
B. u  v  ( v x u )
 
  

C.  (u  v )  (u )  v  ux(v )
 
 
 
D. u  (v  w)  (u  v )  (u  w)
 


E. u  v is orthogonal to both u and v


 
 
F. u  v  u v sin  where  is the angle between u and v


 
G. u  v is the area of a parallelogram having u and v as adjacent sides.
  


H. u and v are parallel if and only if u  v  0
Find the value of each determinant.
3 2
1.
5 7
2 1 3
2.  3  2 1
1
3 2


Let u  i  2 j  3k , v  4i  5 j  6k and w  7i  8 j  9k . Calculate:
 
 
1. u  v
2. u  2v
  
3. v  (u  w)
  
4. u  (v  v )


5. Find a vector orthogonal to both u and v
6. Find the area of the parallelogram with vertices P1 , P2 , P3 , P4 .
P1  (0,0,0), P2  (3,2,1), P3  (1,3,1), P4  (2,1,0)
Vector and Parametric Equations
Worksheet: 1-15 odd
I. Vector Equation
( x, y)  ( x0 , y0 )  t (a, b)
A. ( x0 , y0 ) is the position of the object at t  0
B. (a, b) = direction vector
= slope
= velocity
a  x1  x0 , b  y1  y2
C. speed = velocity  a, b
II. Parametric Equations
If ( x, y)  ( x0 , y0 )  t (a, b) , then x  x0  at and y  y0  bt
III. The Cartesian Equation of a Plane
If ( a, b, c ) is a nonzero vector perpendicular to a plane at ( x0 , y0 , z0 ) , then the equation of the plane is
ax  by  cz  d where d  ax0  by0  cz0
Find the vector and parametric equations for each line.
1. The line through (7, 4) and direction vector (3, -1)
2. The line through (2, 0) and (4, -3)
3. The horizontal line through ( 3 , 5 )
4. A point moves in the plane so that its position P(x,y) at time t is given by the specified vector equation.
(x,y) = (1,2) + t(-3,2)
a) Graph the point’s position at the time t = 0, 1, 2
b) Find the velocity and speed of the moving point.
c) Find the parametric equations of the moving point
5. Find vector and parametric equations of the moving object described.
Velocity = (1, -2) and position at time t=0 is (3, 5)
6. A line has a vector equation (x, y) = (5, 3) + t(-3, 4). Give a pair of parametric equations and a cartesian
equation of the line.
7. Vector (3, 4, -2) is perpendicular to a plane that contains A(0, 1, 2). Find an e