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Calculus > Vectors > Flashcards

Vectors Flashcards

1
Q

Vectors

A

Directional arrows that describe an objects motion and magnitude

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2
Q

Naming vectors

A

[OriginPoint][EndPoint]
Line over head

ie PQ

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3
Q

Scalar-vector multiplication

A

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

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4
Q

Scalar multiple

A

A vector whose magnitude is multiplied by a contant

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5
Q

Parallel vectors

A

Vectors that are scalar multiples of eachother

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6
Q

Equal vectors

A

Vectors of the same direction and magnetude

Not necessarily the same position

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7
Q

Zero vector

A

A vector component with no length

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8
Q

Vector addition

A

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

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9
Q

Vector subtraction

A

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

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10
Q

Position vectors

A

Vector with its head at (0,0)

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11
Q

Standard position

A

(0,0)

The general starting point for object motion

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12
Q

X-component

A

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

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13
Q

Y-component

A

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

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14
Q

Z-component

A

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

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15
Q

Vector magnetude

A

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]

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16
Q

Writing vector coordinates

A

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

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17
Q

Vector head

A

End-point of the vector

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18
Q

Origin point

A

Point at which the vector begins

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19
Q

Unit vector

A

Any vector with a magnitude of 1

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20
Q

Right hand coordinate system

A

Graph ‘z’ up, ‘y’ right, and ‘x’ diagonal towards you

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21
Q

Vector magnetude 3D coordinate plane

A

r=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]

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22
Q

Dot product

A

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)

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23
Q

Orthogonal vectors

A

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

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24
Q

Projections

A

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

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25
Q

Work

A

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

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26
Q

Cross product

A

|u x v|=|u||v|sinθ

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27
Q

Matrix representation of cross product

A

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

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28
Q

Torque

A

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

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29
Q

Vector-valued functions

A

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

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30
Q

Limits of vector-valued functions

A

Lim t→a v(t)=[f(t)]i+[g(t)j+[h(t)]k

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31
Q

Tangent vector

A

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

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32
Q

Unit Tangent Vector

A 
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
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33
Q

Integral of a vector valued function

A 
When v(t)=[f(t)]i+[g(t)j+[h(t)]k,
∫v(t)= [∫f(t)]i + [∫g(t)j + [∫h(t)]k
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34
Q

Uniform motion

A

Motion in a straight line

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35
Q

Arc Length for Vector functions

A

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

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36
Q

Curvature

A

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

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37
Q

Velocity given a curve of a constant radius

A

Always orthogonal

r⊙v=0

38
Q

Curvature Formula

A

K=|v” x v’|/(|v|^3)

39
Q

Principal Unit Normal Vector

A

Given as N=(dT/ds)/(|dT/ds|)

Where T= ∂v/|∂v|

40
Q

Acceleration Components

A

a=(a1)N+(a2)T

41
Q

Equation for a plane through a vector and point

A

d=ax+by+cz

42
Q

Parallel planes

A

Their normal vectors are scalar multiples of eachother

43
Q

Orthogonal plane

A

Normal vectors of which are zero

44
Q

Cylinder

A

A surface containing of all lines parallel to eachother

45
Q

Trace

A

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

46
Q

Xy trace

A

A trace on a plane paralel to z-axis

47
Q

Xz-trace

A

A trace on a plane parallel to y-axis

48
Q

Yz-trace

A

A trace on a plane parallel to x-axis

49
Q

Basic Ellipsoid formula

A

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

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

50
Q

Elliptic parabola

A

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

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

51
Q

One-sheet Hyperboliod

A

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

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

52
Q

Two-sheet hyperboliod

A

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

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

53
Q

Elliptic Code

A

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

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

54
Q

Hyperbolic paraboloid

A

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

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

55
Q

Equations of a line through a point at a specific vector

A

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

56
Q

Normal vector

A

Vector that determines the orientation of a plane

57
Q

Limit of a Vector-Valued Function

A 
Lim r(t)=L
Where r(t) is the vector valued function
58
Q

Vectors may represent

A

Velocity, momentum, acceleration, force, or coordinates

59
Q

Converting velocity vectors to momentum vectors

A

Multiply each i j k component by mass

60
Q

Converting acceleration vectors to momentum vectors

A

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
Q

Converting force vectors to momentum vectors

A

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
Q

Converting directional coordinate functions to momentum vectors

A

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
Q

Converting momentum vectors to velocity vectors

A

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

64
Q

Converting coordinate functions to velocity vectors

A

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

65
Q

Converting force vectors into velocity function vectors

A

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
Q

Converting acceleration vectors into velocity function vectors

A

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

67
Q

Converting velocity vectors to coordinate functions of time

A

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

68
Q

Converting acceleration vectors to coordinate functions of time

A

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
Q

Converting momentum vectors to coordinate functions of time

A

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
Q

Converting force vectors to coordinate functions of time

A

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
Q

Converting force vectors to acceleration vectors

A

Divide each vector component magnitude by object mass

72
Q

Converting velocity function vectors to acceleration vectors

A

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

73
Q

Converting momentum function vectors to acceleration vectors

A

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

74
Q

Converting coordinate function vectors to acceleration vectors

A

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
Q

Converting acceleration vectors to force vectors

A

Multiply by mass for each of the ijk vector components

76
Q

Converting velocity functions to force vectors

A

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

77
Q

Converting momentum functions to force vectors

A

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
Q

Converting coordinate functions to force vectors

A

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
Q

Scalar multiplication

A

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

80
Q

Vector addition

A

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
Q

Vector subtraction

A

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
Q

Cross product

A

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
Q

Dot product

A

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
Q

Scalar triple product

A

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

85
Q

Order of vector opperstions

A

Cross product always comes before dot product

86
Q

Vector triple product

A

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

87
Q

Vector Gradient

A

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
Q

Vector curl

A

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

89
Q

Vector divergence

A

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

90
Q

Variables in vectors

A

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

Calculus (17 decks)

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