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Level 436

Vectors II


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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