Flashcards in Vectors Deck (90):

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## Variables in vectors

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

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## Vectors

### Directional arrows that describe an objects motion and magnitude

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## Naming vectors

###
[OriginPoint][EndPoint]

Line over head

ie PQ

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## Scalar-vector multiplication

###
Multiplying the vector magnitude by a constant (c)

Endpoint (x,y,z) becomes

(cx,cy,cz)

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## Scalar multiple

### A vector whose magnitude is multiplied by a contant

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## Parallel vectors

### Vectors that are scalar multiples of eachother

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## Equal vectors

###
Vectors of the same direction and magnetude

(Not necessarily the same position)

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## Zero vector

### A vector component with no length

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## 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

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## 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

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## Position vectors

### Vector with its head at (0,0)

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## Standard position

###
(0,0)

The general starting point for object motion

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## X-component

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

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## Y-component

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

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## Z-component

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

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## 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]

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## Writing vector coordinates

###

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

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## Vector head

### End-point of the vector

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## Origin point

### Point at which the vector begins

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## Unit vector

### Any vector with a magnitude of 1

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## Right hand coordinate system

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

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## Vector magnetude 3D coordinate plane

### r=√[(x-xi)^2+(y-yi)^2+(z-zi)^2]

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## 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)

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## Orthogonal vectors

###
Vectors whose dot product is zero

u⊙v=0

Yields perpendicular vectors

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## Projections

###
Proj=|u|*cosθ(v/|v|)

Also equals

v*[(u⊙v)/(v⊙v)]

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## 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

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## Cross product

### |u x v|=|u|*|v|*sinθ

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## 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

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## 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

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## 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

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## Limits of vector-valued functions

### Lim t→a v(t)=[f(t)]i+[g(t)j+[h(t)]k

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## 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

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##
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

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## 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

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## Uniform motion

### Motion in a straight line

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## Arc Length for Vector functions

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

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## 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

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## Velocity given a curve of a constant radius

###
Always orthogonal

r⊙v=0

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## Curvature Formula

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

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## Principal Unit Normal Vector

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

Where T= ∂v/|∂v|

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## Acceleration Components

### a=(a1)N+(a2)T

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##
Equation for a plane through a vector and point

### d=ax+by+cz

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## Parallel planes

### Their normal vectors are scalar multiples of eachother

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## Orthogonal plane

### Normal vectors of which are zero

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## Cylinder

### A surface containing of all lines parallel to eachother

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## Trace

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

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## Xy trace

### A trace on a plane paralel to z-axis

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## Xz-trace

### A trace on a plane parallel to y-axis

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## Yz-trace

### A trace on a plane parallel to x-axis

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## 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

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## Elliptic parabola

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

a- ±x-vertices

b- ±y-vertices

c- ±z-vertices

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## 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

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## 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

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## Elliptic Code

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

a- ±x-vertices

b- ±y-vertices

c- ±z-vertices

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## Hyperbolic paraboloid

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

a- ±x-vertices

b- ±y-vertices

c- ±z-vertices

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## Equations of a line through a point at a specific vector

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

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## Normal vector

### Vector that determines the orientation of a plane

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## Limit of a Vector-Valued Function

###
Lim r(t)=L

Where r(t) is the vector valued function

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## Vectors may represent

### Velocity, momentum, acceleration, force, or coordinates

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## Converting velocity vectors to momentum vectors

### Multiply each i j k component by mass

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## 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)

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## 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)

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## 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

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## Converting momentum vectors to velocity vectors

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

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## Converting coordinate functions to velocity vectors

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

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## 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

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## Converting acceleration vectors into velocity function vectors

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

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## 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

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## 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

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## 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

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## 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

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## Converting force vectors to acceleration vectors

### Divide each vector component magnitude by object mass

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## Converting velocity function vectors to acceleration vectors

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

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## 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

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## 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,

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## Converting acceleration vectors to force vectors

### Multiply by mass for each of the ijk vector components

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## Converting velocity functions to force vectors

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

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## 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

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## 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,

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## Scalar multiplication

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

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## 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

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## 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

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## 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

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## 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

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## Scalar triple product

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

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## Order of vector opperstions

### Cross product always comes before dot product

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## Vector triple product

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

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## 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)

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## Vector curl

###
Describes a change in vector direction in terms of its projection onto the other tangent lines

Look it up and practice

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