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List 4 other polar coordinates for P(r,Theta) where
r = 2; Theta = pi / 3 or (2, pi/3).
- 1. (2, pi/3 +2pi) = (2, 7pi/3)
- 2. (2,pi/3-2pi) = (2,-5pi/3)
- 3. (-2, pi/3+pi) = (-2, 4pi/3)
- 4. (-2, pi/3- pi) = (-2, -2pi/3)
What is the polar equation for x?
x = r cos Theta
What is the polar equation for y?
y = r sin Theta
What is the polar equation for y / x?
y/x = tan Theta (where x cannot = 0)
What is the rectangular equation for r2?
r2 = x2 + y2
What are all the values given r = 2 sin Theta for the values of Theta being:
0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, and pi
What does a cardioid look like? What is the equation for a cardioid?
- r = a +or - b cos theta
- r = a +or- b sin theta
What does symmetry about the y axis look like?
What does a Limacons look like?
- a > b ; a = b a < b
- The bigger the constant in front of sin gets, the bigger the loop gets inside the circle.
What does Lemniscates look like?
- r2 = a2 cos (2 Theta)
- r2 = a2 sin(2 Theta)
What is the equation for a rose?
- r = a cos n Theta
- n = even #, then there will be 2n number of leaves.
- n = odd #, then there will be n number of leaves.
What is the polar form of a + bi?
z = r ( cos theta + i sin theta)
What is DeMoivre's theorem for :
z = r ( cos theta + i sin theta)?
zn = r n( cos theta + i sin theta)
What is the nth roots of complex numbers for:
z = r (cos theta + i sin theta)?
- If n is a positive intiger, than:
- wk = r1/n [cos (theta + 2kpi)/n + i sin (theta +2kpi)/n]
- for k = 0, 1, 2, 3. . . . . n-1.
What are the general formulas for adding, subtracting and multiplying by a constant to vectors? given:
u = <a1,b1> and v = <a2,b2>
- u + v = <a1+a2,b1+b2>
- u - v = <a1-a2,b1-b2>
- cu = <ca1,cb1>, c = real number
Addition Properties of Vectors.
- u + v = v + u
- u + (v + w ) = ( u + v ) + w
- u + 0 = u
- u + (-u) = 0
Length Propertie for Vectors
| cu | = | c| |u |
Multiplication Properties of Vectors.
- c (u + v ) = cu + cv
- ( c + d ) u = cu + du
- (cd) u = c (du) = d (cu)
- 1u = u
- 0u = 0
- c 0 = 0
How is x represented in vector form?
x = i = < 1 , 0 >
How is y represented in vector form?
y = j = < 0, 1 >
Horizontal and vertical components of a vector.
- magnitude = | v | in the direction of theta.
- Then v = < a, b > = ai + bj, where
- a = | v | cos theta i and b = | v | sin theta j
- thus, we can express v as:
- v = | v | cos theta i + | v | sin theta j
Given u = < a1 , b1 >
v = < a2 , b2 >
u * v = a1a2 + b1b2
Properties of dot product.
- u * v = v * u
- (au) * v = a (u*v) = u * (av)
- ( u + v) * w = u * w + v * w
- | u |2 = u * u
The dot product theorem
u * v = | u | |v | cos theta
Angle between two vectors
cos theta = ( u * v) / [ | u| | v | ]
- two nonzero vecters are perpendicular when their dot product is 0:
- u * v = 0
The component u along v is:
( u * v ) / | v |
The projection of u onto v is the vector projv u given by
- projv u = [ ( u *v ) / | v |2 ] v
- If the vector u is resolved into u1 and u2, where u1 is parallel to v and u2 is orthagonal to v, then
- u1 = projvu and u2 = u - projvu
- W = F * D
- W= work
- F= force
- D= vector