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|=âˆš[(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=âˆš[(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

uâŠ™v=|u|*|v|*cosÎ¸=(ux)(vx)+(uy)(vy)+(uz)(vz)

23

## Orthogonal vectors

###
Vectors whose dot product is zero

uâŠ™v=0

Yields perpendicular vectors

24

## Projections

###
Proj=|u|*cosÎ¸(v/|v|)

Also equals

v*[(uâŠ™v)/(vâŠ™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=FâŠ™d=|F|*|d|*cosÎ¸

26

## Cross product

### |u x v|=|u|*|v|*sinÎ¸

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 tâ†’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,

âˆ«v(t)= [âˆ«f(t)]i + [âˆ«g(t)j + [âˆ«h(t)]k

34

## Uniform motion

### Motion in a straight line

35

## Arc Length for Vector functions

### L=âˆ«âˆš[f'(t)^2+g'(t)^2+h'(t)^2]dt = âˆ«|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= âˆ‚v/|âˆ‚v|

And 's' equals arc length over a set interval

37

## Velocity given a curve of a constant radius

###
Always orthogonal

râŠ™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= âˆ‚v/|âˆ‚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- Â±x-vertices

b- Â±y-vertices

c- Â±z-vertices

50

## Elliptic parabola

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

a- Â±x-vertices

b- Â±y-vertices

c- Â±z-vertices

51

## One-sheet Hyperboliod

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

a- Â±x-vertices

b- Â±y-vertices

c- Â±z-vertices

52

## Two-sheet hyperboliod

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

a- Â±x-vertices

b- Â±y-vertices

c- Â±z-vertices

53

## Elliptic Code

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

a- Â±x-vertices

b- Â±y-vertices

c- Â±z-vertices

54

## Hyperbolic paraboloid

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

a- Â±x-vertices

b- Â±y-vertices

c- Â±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 âˆ«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 âˆ«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 âˆ‚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 âˆ‚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 âˆ«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 âˆ«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 âˆ«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 âˆ«a(t)+Vo=v(t) and then âˆ«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 âˆ«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 âˆ«a(t)+Vo=v(t) and then âˆ«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 âˆ‚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 âˆ‚v(t)=a for each i j k component

74

## Converting coordinate function vectors to acceleration vectors

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

78

## Converting coordinate functions to force vectors

### Take the derivative âˆ‚s(t)=v for each i j k component, then take the derivative âˆ‚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