Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

► Calculus > Vectors > Flashcards

Vectors Flashcards

1
Q

Vectors

A

Directional arrows that describe an objects motion and magnitude

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Naming vectors

A

[OriginPoint][EndPoint]
Line over head

ie PQ

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Scalar-vector multiplication

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Scalar multiple

A

A vector whose magnitude is multiplied by a contant

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Parallel vectors

A

Vectors that are scalar multiples of eachother

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Equal vectors

A

Vectors of the same direction and magnetude

Not necessarily the same position

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Zero vector

A

A vector component with no length

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Position vectors

A

Vector with its head at (0,0)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Standard position

A

(0,0)

The general starting point for object motion

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Writing vector coordinates

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Vector head

A

End-point of the vector

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Origin point

A

Point at which the vector begins

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Unit vector

A

Any vector with a magnitude of 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Right hand coordinate system

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Vector magnetude 3D coordinate plane

A

r=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Orthogonal vectors

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Projections

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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)

► Calculus (15 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2025 Bold Learning Solutions. Terms and Conditions