Level 435
Level 437

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Scalar Product

a.b = |a||b|cosθ

Scalar Product is

commutative and associative

a.a

|a|²

a.b = 0

vectors a and b are perpendicular

Vector Product

a ^ b = |a||b|sinθ n

a ^ b

- b ^ a

|a ^ b|

Area of parallelogram formed by a and b

a ^ b = 0

a and b are parallel

0

a ^ a

a . (a ^ b)

0 since a ^ b is perpendicular to a

Vector Product is

Anti-commutative, distributive and non-associative

Vector Area of a plane surface

Vector of magnitude equal to the area, direction which is normal to the surface

Vector Area of a parallelogram

Given by S = a ^ b

Given by A = S.z

Area projected onto the x-y plane

Vector Area of non-planar surface

Vector sum of individual vector areas

Vector Area of a closed surface

0, with all vector areas outwards

Scalar Triple Product

[a,b,c] = a.(b ^ c)

Scalar Triple Product is

invariant under cyclic permutations

Scalar Triple Product [a,b,c]

equal to the volume of the parallelepiped formed by a, b and c

Scalar Product a.b

equal to the product of the length of a and the projection of b onto a (given by bcosθ)

Vector Triple Product

a ^ (b ^ c) = (a.c)b - (a.b)c

Vector Straight Line

r = a + λl or r ^ l = a ^ l

Length of line

Given by λ when l is a unit vector

Vector equation of line through two points

r = a + λ(b - a)

Vector Equation of a plane

r = a + λp + µq or r.n = d (where d is the distance from the origin)

Vector Equation of a plane containing points a,b,c

r = a + λ(b - a) + µ(c - a)

constitute a basis

Any three non-coplanar vectors

v = λa + µb + νc

Vector written in terms of components of basis

Orthogonal Basis

have mutually perpendicular vectors

[i,j,k] = 1

Right Handed Orthonormal Basis

its own reciprocal

Right Handed Orthonormal Basis is

a.b =

axbx + ayby + azbz

a ^ b =

(aybz - azby)i + (azbx - axbz)j + (axby - aybx)k

rcosø

Cylindrical Polar Coordinates x

rsinø

Cylindrical Polar Coordinates y

z

Cylindrical Polar Coordinates z

r drdødz

Cylindrical Polar Coordinates Volume Element

rsinθcosø

Spherical Polar Coordinates x

rsinθsinø

Spherical Polar Coordinates y

rcosθ

Spherical Polar Coordinates z

r²sinθ drdθdø

Spherical Polar Coordinates Volume Element

|(a+b)|²

(a+b) . (a+b)

vector

the direction or course followed by an airplane, missile, or the like

Magnitude

The number of degrees a regular polygon can be rotated to map onto itself.

direction

a positive ________ or association means that, in general, as one variable increases, so does the other; when increases in one variable generally correspond to decreases in the other, the association is negative

component form

<a,b> to find: <x₂-x₁,y₂-x₁>

u+v=<u₁,+v₁,u₁+v₂,v₁>

vector addition

xu=x<u₁,u₂>=<xu₁,xu₂>

multiplying a number and vector

u=v/|v|=1/|v|(v)

unit vector

ai+bj

<a,b> in i j form

v=<|v|cosθ,|v|sinθ>

components of v using direction angle

direction angle

x=|u|cosθ or tanθ=y/x

u°v=u₁v₁+u₂v₂

dot product

orthagonal

u°v=0

cosθ=(u°v/|u||v|)

the angle between

a=p+w

plane problems

p

air speed

a

ground speen

(r,θ+2nπ) or (-r,θ+(2n+1)π)

to find all polar coordinates

(r, θ)

polar coordinates

r²=x²+y² and tanθ=y/x

switch from rectangular to polar

x=rcosθ or y=rsinθ

switch from polar to rectangular