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Flashcards in Vectors Deck (90):
1

Vectors

Directional arrows that describe an objects motion and magnitude

2

Naming vectors

[OriginPoint][EndPoint]
Line over head

ie PQ

3

Scalar-vector multiplication

Multiplying the vector magnitude by a constant (c)
Endpoint (x,y,z) becomes
(cx,cy,cz)

4

Scalar multiple

A vector whose magnitude is multiplied by a contant

5

Parallel vectors

Vectors that are scalar multiples of eachother

6

Equal vectors

Vectors of the same direction and magnetude
(Not necessarily the same position)

7

Zero vector

A vector component with no length

8

Vector addition

u+v
Reposition the vectors 'u' and 'v' so that they form two sides of a triangle
Use c^2=a^2+b^2, one at a time for multiple vectors
Add endpoint vx to ux, vy to uy, and vz to uz

9

Vector subtraction

u-v
Change the direction of vector 'v'
Reposition the vectors 'u' and new 'v' so that they form two sides of a triangle
Use c^2=a^2+b^2
Subtract endpoint vx from ux, vy from uy, and vz from uz

10

Position vectors

Vector with its head at (0,0)

11

Standard position

(0,0)
The general starting point for object motion

12

X-component

The vector that measures how far the object moves in simply the x-direction, if y and z were ignored

13

Y-component

The vector that measures how far the object moves in simply the y-direction, if x and z were ignored

14

Z-component

The vector that measures how far the object moves in simply the z-direction, if y and x were ignored

15

Vector magnetude

The 'length' of the vector, typically a representation of acceleration
Denoted |PQ|
Use Pythagorean theorem
|PQ|=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]

16

Writing vector coordinates


Where u1is the origin point for the vector and u2 is the vector head

17

Vector head

End-point of the vector

18

Origin point

Point at which the vector begins

19

Unit vector

Any vector with a magnitude of 1

20

Right hand coordinate system

Graph 'z' up, 'y' right, and 'x' diagonal towards you

21

Vector magnetude 3D coordinate plane

r=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]

22

Dot product

Product of the magnitudes of each vector and the 'cos' of the angle between the two vectors, is equal to the sum of their component products
u⊙v=|u|*|v|*cosθ=(ux)(vx)+(uy)(vy)+(uz)(vz)

23

Orthogonal vectors

Vectors whose dot product is zero
u⊙v=0
Yields perpendicular vectors

24

Projections

Proj=|u|*cosθ(v/|v|)
Also equals
v*[(u⊙v)/(v⊙v)]

25

Work

Dot product of the force vector and the traveling vector (in which the object actually travels, gives the angle measure if you have the magnitudes
W=F⊙d=|F|*|d|*cosθ

26

Cross product

|u x v|=|u|*|v|*sinθ

27

Matrix representation of cross product

i. j. k.
u. = u x v =|j and k|i+|i and k|j+|j and i|k
v

28

Torque

Cross product of force and radius, outward pressure due to force on a plane, think of tightening a wrench (torque is the force pushing a screw in or out of a hole)
r x F

29

Vector-valued functions

Expressing an algebraic curve in space with three functions i,j,k
v(t)=[f(t)]i+[g(t)j+[h(t)]k

30

Limits of vector-valued functions

Lim t→a v(t)=[f(t)]i+[g(t)j+[h(t)]k

31

Tangent vector

Basically the derivative if the vector valued function
When v(t)=[f(t)]i+[g(t)j+[h(t)]k,
Then
v'(t)= [f'(t)]i + [g'(t)j + [h'(t)]k

32

Unit Tangent Vector

Gives the direction but not the length of the tangent vector at point-t
T(t)=v'(t)/|v'(t)|
For v(t)= [f(t)]i + [g(t)j + [h(t)]k

33

Integral of a vector valued function

When v(t)=[f(t)]i+[g(t)j+[h(t)]k,
∫v(t)= [∫f(t)]i + [∫g(t)j + [∫h(t)]k

34

Uniform motion

Motion in a straight line

35

Arc Length for Vector functions

L=∫√[f'(t)^2+g'(t)^2+h'(t)^2]dt = ∫|v'(t)|dt

36

Curvature

A measure of how quickly the direction of the curve changes over a given interval
Given as K=|dT/ds|
Where T= ∂v/|∂v|
And 's' equals arc length over a set interval

37

Velocity given a curve of a constant radius

Always orthogonal
r⊙v=0

38

Curvature Formula

K=|v" x v'|/(|v|^3)

39

Principal Unit Normal Vector

Given as N=(dT/ds)/(|dT/ds|)
Where T= ∂v/|∂v|

40

Acceleration Components

a=(a1)N+(a2)T

41

Equation for a plane through a vector and point

d=ax+by+cz

42

Parallel planes

Their normal vectors are scalar multiples of eachother

43

Orthogonal plane

Normal vectors of which are zero

44

Cylinder

A surface containing of all lines parallel to eachother

45

Trace

Set of points at which a surface intersects a plane parallel to either x,y, or z coordinate planes

46

Xy trace

A trace on a plane paralel to z-axis

47

Xz-trace

A trace on a plane parallel to y-axis

48

Yz-trace

A trace on a plane parallel to x-axis

49

Basic Ellipsoid formula

(x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

50

Elliptic parabola

(x^2)/(a^2)+(y^2)/(b^2)=z

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

51

One-sheet Hyperboliod

(x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

52

Two-sheet hyperboliod

-(x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

53

Elliptic Code

(x^2)/(a^2)+(y^2)/(b^2)=(z^2)/(c^2)

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

54

Hyperbolic paraboloid

(x^2)/(a^2)-(y^2)/(b^2)=z

a- ±x-vertices
b- ±y-vertices
c- ±z-vertices

55

Equations of a line through a point at a specific vector

(X,y,z)=(z0,y0,x0)+t(a,b,c)

56

Normal vector

Vector that determines the orientation of a plane

57

Limit of a Vector-Valued Function

Lim r(t)=L
Where r(t) is the vector valued function

58

Vectors may represent

Velocity, momentum, acceleration, force, or coordinates

59

Converting velocity vectors to momentum vectors

Multiply each i j k component by mass

60

Converting acceleration vectors to momentum vectors

Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions then multiply by mass to find each component-function p(t)=m*v(t)

61

Converting force vectors to momentum vectors

Divide each force i j k component by object mass to find acceleration components. Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions then multiply by mass to find each component-function p(t)=m*v(t)

62

Converting directional coordinate functions to momentum vectors

Take the derivative ∂s(t)=v for each i j k component, multiply by object mass to find p=m*v in each component direction

63

Converting momentum vectors to velocity vectors

Divide the momentum vector for each of the i j k components by object mass

64

Converting coordinate functions to velocity vectors

Take the derivative ∂s(t)=v for each i j k component,

65

Converting force vectors into velocity function vectors

Divide each force i j k component by object mass to find acceleration components then calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions

66

Converting acceleration vectors into velocity function vectors

Calculate the indefinate integral ∫a(t)+Vo=v(t) for each of the i j k directions

67

Converting velocity vectors to coordinate functions of time

Calculate the indefinate integral ∫v(t)+So=s(t) for each of the i j k directions

68

Converting acceleration vectors to coordinate functions of time

Calculate the indefinate integral ∫a(t)+Vo=v(t) and then ∫v(t)+So=s(t) for each of the i j k directions

69

Converting momentum vectors to coordinate functions of time

Divide each of the i j k components by object mass then calculate the indefinate integral ∫v(t)+So=s(t) for each of the x y z directions

70

Converting force vectors to coordinate functions of time

Divide each i j k component by mass for acceleration vectors then calculate the indefinate integral ∫a(t)+Vo=v(t) and then ∫v(t)+So=s(t) for each of the x y z directions

71

Converting force vectors to acceleration vectors

Divide each vector component magnitude by object mass

72

Converting velocity function vectors to acceleration vectors

Take the derivative ∂v(t)=a for each i j k component,

73

Converting momentum function vectors to acceleration vectors

Divide each i j k component by object mass then take the derivative ∂v(t)=a for each i j k component

74

Converting coordinate function vectors to acceleration vectors

Take the derivative ∂s(t)=v for each i j k component, then take the derivative ∂v(t)=a for each i j k component,

75

Converting acceleration vectors to force vectors

Multiply by mass for each of the ijk vector components

76

Converting velocity functions to force vectors

Take the derivative ∂v(t)=a and multiply by object mass for each i j k component

77

Converting momentum functions to force vectors

Divide each i j k component by object mass then take the derivative ∂v(t)=a and multiply by mass for each i j k component

78

Converting coordinate functions to force vectors

Take the derivative ∂s(t)=v for each i j k component, then take the derivative ∂v(t)=a and multiply by mass for each i j k component,

79

Scalar multiplication

The magnetude of the vector is changed by multiplying a constant to all directional components

80

Vector addition

If you look at the two vectors as the legs of a triangle (changing their position but not their diection or magnetude)- tail to head, then the final vector would be the third leg of that triangle

81

Vector subtraction

If you look at the two vectors as the legs of a triangle (changing their position but not their diection or magnetude)- tail to tail, then the final vector would be the third leg of that triangle

82

Cross product

When the two vectors are written one on top of the other in a matrix
The determinant of the vertical-submatrix for each direction
Always just a vector, of moment for those two vectors

83

Dot product

When the two vectors are written one on top of the other in a matrix
The sum of the determinants of the vertical-submatrix for each direction
Always just a number

84

Scalar triple product

The dot product of a vector and the cross product of two others

85

Order of vector opperstions

Cross product always comes before dot product

86

Vector triple product

The cross product of a vector and the cross product of two other vectors

87

Vector Gradient

Sum of the partial derivatives in each of those vector field directions
Measures the rate and direction of change in a scalar field, points to greatest potential, where the tangent is zero - lowest point for gravitational potential, highest voltage point for electrical-field potential, most positively charged point for magnetic field potential; where the ball/electron will most likely 'roll' (accelerate)

88

Vector curl

Describes a change in vector direction in terms of its projection onto the other tangent lines
Look it up and practice

89

Vector divergence

Describes the 3ddirection inwhich an object will tend to move
Limit of the Double integral of a projection

90

Variables in vectors

(x,y,z)=(i,j,k)