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MathFormulas 2020 (Get A Good Start Course)

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Math Formulas
Percentage and interest rate
𝐾𝐾 = 𝐾𝐾0 · (1 + π‘Ÿπ‘Ÿ)𝑛𝑛
𝐾𝐾0 : Initial value
n: number of interest terms
𝐾𝐾: Future value (after n interest terms)
r: interest rate (growth rate)
π‘Žπ‘Ž = 1 + π‘Ÿπ‘Ÿ
a: growth factor.
Factorization and Quadratic Identities
π‘₯π‘₯ · π‘Žπ‘Ž + π‘₯π‘₯ · 𝑏𝑏 = π‘₯π‘₯ · (π‘Žπ‘Ž + 𝑏𝑏)
(π‘Žπ‘Ž ± 𝑏𝑏)2 = π‘Žπ‘Ž2 + 𝑏𝑏 2 ± 2π‘Žπ‘Ž · 𝑏𝑏
Fraction rules
Exponent rules
π‘Žπ‘Ž2 − 𝑏𝑏 2 = (π‘Žπ‘Ž − 𝑏𝑏) · (π‘Žπ‘Ž + 𝑏𝑏)
𝑏𝑏
𝑐𝑐
π‘Žπ‘Ž · =
π‘Žπ‘Ž·π‘π‘
𝑐𝑐
π‘Žπ‘Ž/𝑏𝑏
𝑐𝑐/𝑑𝑑
=
π‘Žπ‘Ž·π‘‘𝑑
𝑏𝑏·π‘π‘
π‘Žπ‘Ž
𝑏𝑏/𝑐𝑐
=
π‘Žπ‘Ž·π‘π‘
𝑏𝑏
π‘Žπ‘Žπ‘Ÿπ‘Ÿ · π‘Žπ‘Ž 𝑠𝑠 = π‘Žπ‘Žπ‘Ÿπ‘Ÿ+𝑠𝑠
π‘Žπ‘Žπ‘Ÿπ‘Ÿ
= π‘Žπ‘Žπ‘Ÿπ‘Ÿ−𝑠𝑠
π‘Žπ‘Ž 𝑠𝑠
(π‘Žπ‘Žπ‘Ÿπ‘Ÿ )𝑠𝑠 = π‘Žπ‘Žπ‘Ÿπ‘Ÿ·π‘ π‘ 
(π‘Žπ‘Ž · 𝑏𝑏)π‘Ÿπ‘Ÿ = π‘Žπ‘Žπ‘Ÿπ‘Ÿ · 𝑏𝑏 π‘Ÿπ‘Ÿ
π‘Žπ‘Ž π‘Ÿπ‘Ÿ π‘Žπ‘Žπ‘Ÿπ‘Ÿ
οΏ½ οΏ½ = π‘Ÿπ‘Ÿ
𝑏𝑏
𝑏𝑏
π‘Žπ‘Ž0 = 1 ,
𝑠𝑠
π‘Žπ‘Ž−π‘Ÿπ‘Ÿ =
π‘Žπ‘Ž ≠ 0
1
π‘Žπ‘Žπ‘Ÿπ‘Ÿ
√π‘Žπ‘Žπ‘Ÿπ‘Ÿ = π‘Žπ‘Ž
π‘Ÿπ‘Ÿ
οΏ½ οΏ½
𝑠𝑠
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1
π‘Žπ‘Ž
𝑏𝑏
𝑐𝑐
𝑑𝑑
· =
π‘Žπ‘Ž·π‘π‘
𝑏𝑏·π‘‘𝑑
π‘Žπ‘Ž/𝑏𝑏
𝑐𝑐
=
π‘Žπ‘Ž
𝑏𝑏·π‘π‘
Logarithm rules
The common logarithm:
log(π‘Žπ‘Ž π‘₯π‘₯ ) = π‘₯π‘₯ · log (π‘Žπ‘Ž)
log(π‘Žπ‘Ž · 𝑏𝑏) = log(π‘Žπ‘Ž) + log(𝑏𝑏)
π‘Žπ‘Ž
log οΏ½ οΏ½ = log(π‘Žπ‘Ž) − log (𝑏𝑏)
𝑏𝑏
log(10) = 1
Similar rules apply to the natural logarithm (and all other logarithm functions)
ln(π‘Žπ‘Ž π‘₯π‘₯ ) = π‘₯π‘₯ · ln (π‘Žπ‘Ž)
ln(π‘Žπ‘Ž · 𝑏𝑏) = ln(π‘Žπ‘Ž) + ln(𝑏𝑏)
π‘Žπ‘Ž
ln οΏ½ οΏ½ = ln(π‘Žπ‘Ž) − ln (𝑏𝑏)
𝑏𝑏
ln(𝑒𝑒) = 1
Graph of the natural logarithm (base e)
Only defined for x>0
Vertical asymptote at x=0
Intersects the x-axis at x=1 ln(1) = 0
Heads towards infinity ln(π‘₯π‘₯) → ∞ for π‘₯π‘₯ → ∞
Intersects the point (e,1)
Graph of the common logarithm (base 10)
Only defined for x>0
Vertical asymptote at x=0
Intersects the x-axis at x=1 log(1) = 0
Heads towards infinity log(π‘₯π‘₯) → ∞ for π‘₯π‘₯ → ∞
Intersects the point (10,1)
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2
Functions, graphs and tangents
Linear function
Definition
𝑓𝑓(π‘₯π‘₯) = π‘Žπ‘Ž · π‘₯π‘₯ + 𝑏𝑏
The graph of a linear function is a straight line:
𝑦𝑦 = π‘Žπ‘Ž · π‘₯π‘₯ + 𝑏𝑏
a: the slope of the line
b: 2nd coordinate of the point where the graph
intersects with the y-axis.
The slope can be calculated by use of two points (π‘₯π‘₯1 , 𝑦𝑦1 ) og (π‘₯π‘₯2 , 𝑦𝑦2 ) on the graph:
π‘Žπ‘Ž =
Δy y2 − y1
=
Δx x2 − x1
The equation of a tangent t to the graph of the
function 𝑓𝑓 at (π‘₯π‘₯0 , 𝑓𝑓(π‘₯π‘₯0 )):
𝑦𝑦 = 𝑓𝑓´(π‘₯π‘₯0 ) · (π‘₯π‘₯ − π‘₯π‘₯0 ) + 𝑓𝑓(π‘₯π‘₯0 )
The equation of a straight line can also be expressed
π‘Žπ‘Ž · (π‘₯π‘₯ − π‘₯π‘₯0 ) + 𝑏𝑏 · (𝑦𝑦 − 𝑦𝑦0 ) = 0
π‘Žπ‘Ž
where οΏ½ οΏ½ is a vector that is perpendicular to the line and (π‘₯π‘₯0 , 𝑦𝑦0 ) is a point on the line.
𝑏𝑏
To fit a linear model to a data set you use linear regression analysis.
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3
Exponential function
Definition:
𝑓𝑓(π‘₯π‘₯) = 𝑏𝑏 · π‘Žπ‘Ž π‘₯π‘₯ , 𝑏𝑏 > 0 , π‘Žπ‘Ž > 0 , π‘₯π‘₯ ∈ ℝ
a: growth factor (a>1)
a: decay factor (0<a<1)
b: second coordinate of the point where
the graph intersects with the y-axis.
π‘Žπ‘Ž = 1 + π‘Ÿπ‘Ÿ
Growth constant
Decay constant
π‘Žπ‘Ž =
𝑇𝑇2 =
𝑇𝑇½ =
π‘₯π‘₯2 −π‘₯π‘₯1
οΏ½
𝑦𝑦2
𝑦𝑦1
log(2) ln(2)
=
log(π‘Žπ‘Ž) ln(π‘Žπ‘Ž)
1
log οΏ½2οΏ½
log(π‘Žπ‘Ž)
=
1
ln οΏ½2οΏ½
ln(π‘Žπ‘Ž)
Exponential growth
A certain change of x produces a certain %-change of the range element regardless of the initial value of x:
f ( x + h) = a h ⋅ f ( x )
To fit an exponential model to a data set you use exponential regression analysis.
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4
Power function
Definition:
𝑓𝑓(π‘₯π‘₯) = 𝑏𝑏 · π‘₯π‘₯ π‘Žπ‘Ž
, 𝑏𝑏 > 0 , π‘Žπ‘Ž ∈ ℝ , π‘₯π‘₯ > 0
Determination of the exponent a using 2 points (π‘₯π‘₯1 , 𝑦𝑦1 ) og (π‘₯π‘₯2 , 𝑦𝑦2 ) on the graph:
π‘Žπ‘Ž =
log(𝑦𝑦2 ) − log(𝑦𝑦1 )
log(π‘₯π‘₯2 ) − log(π‘₯π‘₯1 )
π‘Žπ‘Ž =
ln(𝑦𝑦2 ) − ln(𝑦𝑦1 )
ln(π‘₯π‘₯2 ) − ln(π‘₯π‘₯1 )
When the value of x increases by a certain percentage the range element of a power function will also
change by a certain percentage regardless of the initial value of x:
f (h ⋅ x) = h a ⋅ f ( x)
To fit a power model to a data set you use power regression analysis.
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5
Polynomial
Degree n polynomial
𝑝𝑝(π‘₯π‘₯) = π‘Žπ‘Žπ‘›π‘› · π‘₯π‘₯ 𝑛𝑛 + π‘Žπ‘Žπ‘›π‘›−1 · π‘₯π‘₯ 𝑛𝑛−1 + β‹― + π‘Žπ‘Ž1 · π‘₯π‘₯ + π‘Žπ‘Ž0
Degree 1 polynomial
Degree 2 polynomial
𝑓𝑓(π‘₯π‘₯) = π‘Žπ‘Ž · π‘₯π‘₯ + 𝑏𝑏
𝑝𝑝(π‘₯π‘₯) = π‘Žπ‘Ž · π‘₯π‘₯ 2 + 𝑏𝑏 · π‘₯π‘₯ + 𝑐𝑐
Roots (= zeros) of the degree 2 polynomial (if any)
π‘₯π‘₯ =
−𝑏𝑏 ± √𝑑𝑑
2π‘Žπ‘Ž
𝑑𝑑 = 𝑏𝑏 2 − 4π‘Žπ‘Ž · 𝑐𝑐
When π‘₯π‘₯1 and π‘₯π‘₯2 are roots (could be alike) in a degree 2 polynomial it can be factorized
𝑝𝑝(π‘₯π‘₯) = π‘Žπ‘Ž(π‘₯π‘₯ − π‘₯π‘₯1 )(π‘₯π‘₯ − π‘₯π‘₯2 )
Any polynomial 𝑝𝑝 with one or more roots can be factorized:
𝑝𝑝(π‘₯π‘₯) = (π‘₯π‘₯ − π‘₯π‘₯1 )π‘žπ‘ž(π‘₯π‘₯)
where π‘₯π‘₯1 is a root of 𝑝𝑝 and q(x) is a polynomial with or without any roots.
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6
Trigonometric functions
The unit circle
The length x is equal to the angle v in radians
π‘₯π‘₯ =
𝑣𝑣 =
Definition of cosine og sine
Degrees
Sin
30°
45°
60°
90°
0
1
2
√2
2
√3
2
1
√3
3
1
√3
-
1
Cos
0
Tan
180
π‘₯π‘₯
πœ‹πœ‹
0°
0
Radians
πœ‹πœ‹
𝑣𝑣
180
πœ‹πœ‹
6
√3
2
πœ‹πœ‹
4
√2
2
πœ‹πœ‹
3
1
2
πœ‹πœ‹
2
0
sin2 (π‘₯π‘₯) + cos2 (π‘₯π‘₯) = 1
Cosine
Cosine is periodic:
Cosine is even:
Sine
Sine is periodic:
Sine is odd:
Harmonic function
7
cos(π‘₯π‘₯ + 2πœ‹πœ‹) = cos (π‘₯π‘₯)
cos(−π‘₯π‘₯) = cos (π‘₯π‘₯)
cos(πœ‹πœ‹ − π‘₯π‘₯) = −cos (π‘₯π‘₯)
sin(π‘₯π‘₯ + 2πœ‹πœ‹) = sin (π‘₯π‘₯)
sin(−π‘₯π‘₯) = −sin (π‘₯π‘₯)
sin(πœ‹πœ‹ − π‘₯π‘₯) = sin (π‘₯π‘₯)
f (t ) = A ⋅ sin(ω ⋅ t + Ο• ) + k
A : amplitude
ω : angular frequency
Ο• : initial phase
k : vertical shift
T=
2π
ω
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Ο•
: phase shift (to the left)
ω
Differential calculus
The derivative of f at x = x 0 :
𝑓𝑓(π‘₯π‘₯0 +β„Ž)−𝑓𝑓(π‘₯π‘₯0 )
β„Ž
β„Ž→0
𝑓𝑓 ′ (π‘₯π‘₯0 ) = lim
Differentiation rules
f ´(x) = lim
x → x0
f ( x) − f ( x 0 )
x − x0
′
οΏ½π‘˜π‘˜ · 𝑓𝑓(π‘₯π‘₯)οΏ½ = π‘˜π‘˜ · 𝑓𝑓 ′ (π‘₯π‘₯)
′
�𝑓𝑓(π‘₯π‘₯) ± 𝑔𝑔(π‘₯π‘₯)οΏ½ = 𝑓𝑓 ′ (π‘₯π‘₯) ± 𝑔𝑔′ (π‘₯π‘₯)
′
�𝑓𝑓(π‘₯π‘₯) · 𝑔𝑔(π‘₯π‘₯)οΏ½ = 𝑓𝑓 ′ (π‘₯π‘₯) · 𝑔𝑔(π‘₯π‘₯) + 𝑓𝑓(π‘₯π‘₯) · 𝑔𝑔′(π‘₯π‘₯)
′
�𝑓𝑓�𝑔𝑔(π‘₯π‘₯)οΏ½οΏ½ = 𝑓𝑓′(𝑔𝑔(π‘₯π‘₯)) · 𝑔𝑔′(π‘₯π‘₯)
Function
The derivative
ln(x)
1
π‘₯π‘₯
𝑒𝑒 π‘₯π‘₯
8
𝑒𝑒 π‘₯π‘₯
𝑒𝑒 π‘˜π‘˜·π‘₯π‘₯
π‘˜π‘˜ · 𝑒𝑒 π‘˜π‘˜·π‘₯π‘₯
π‘Žπ‘Ž π‘₯π‘₯
π‘Žπ‘Ž π‘₯π‘₯ · ln (π‘Žπ‘Ž)
π‘₯π‘₯ π‘Žπ‘Ž
π‘Žπ‘Ž · π‘₯π‘₯ π‘Žπ‘Ž−1
√π‘₯π‘₯
1
2√π‘₯π‘₯
1
π‘₯π‘₯
−
cos (π‘₯π‘₯)
sin (π‘₯π‘₯)
k
(constant)
1
π‘₯π‘₯ 2
− sin(π‘₯π‘₯)
cos(π‘₯π‘₯)
0
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Integral calculus
Function
Indefinite integral (antiderivative) (+ an arbitrary constant)
𝑒𝑒 π‘₯π‘₯
𝑒𝑒 π‘₯π‘₯
𝑒𝑒 π‘˜π‘˜·π‘₯π‘₯
1
π‘˜π‘˜
π‘₯π‘₯ π‘Žπ‘Ž
1
π‘₯π‘₯ π‘Žπ‘Ž+1
π‘Žπ‘Ž+1
√π‘₯π‘₯
2
π‘₯π‘₯√π‘₯π‘₯
3
π‘Žπ‘Ž π‘₯π‘₯
1
π‘₯π‘₯
cos (π‘₯π‘₯)
sin (π‘₯π‘₯)
k (constant)
· 𝑒𝑒 π‘˜π‘˜·π‘₯π‘₯
π‘Žπ‘Ž π‘₯π‘₯
ln (π‘Žπ‘Ž)
ln|x|
sin (π‘₯π‘₯)
−cos (π‘₯π‘₯)
k· π‘₯π‘₯
Integration rules
9
F (x) is an antiderivative of f
Indefinite integrals
∫ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = 𝐹𝐹(π‘₯π‘₯) + 𝑐𝑐
∫ π‘˜π‘˜ · 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = π‘˜π‘˜ · ∫ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
∫ 𝑓𝑓(π‘₯π‘₯) ± 𝑔𝑔(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = ∫ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 ± ∫ 𝑔𝑔(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
Integration by substitution:
∫ 𝑓𝑓(𝑔𝑔(π‘₯π‘₯)) · 𝑔𝑔′(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = ∫ 𝑓𝑓(𝑑𝑑)𝑑𝑑𝑑𝑑
Integration by parts:
∫ f ( x) ⋅ g ( x)dx = F ( x) ⋅ g ( x) − ∫ F ( x) ⋅ g´(x)dx
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, t = g (x)
𝑏𝑏
∫π‘Žπ‘Ž 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = [𝐹𝐹(π‘₯π‘₯)]π‘π‘π‘Žπ‘Ž = 𝐹𝐹(𝑏𝑏) − 𝐹𝐹(π‘Žπ‘Ž)
Definite integrals:
𝑏𝑏
𝑐𝑐
𝑏𝑏
οΏ½ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = οΏ½ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 + οΏ½ 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
π‘Žπ‘Ž
𝑏𝑏
π‘Žπ‘Ž
𝑐𝑐
𝑏𝑏
∫π‘Žπ‘Ž π‘˜π‘˜ · 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = π‘˜π‘˜ · ∫π‘Žπ‘Ž 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
𝑏𝑏
𝑏𝑏
𝑏𝑏
∫π‘Žπ‘Ž 𝑓𝑓(π‘₯π‘₯) ± 𝑔𝑔(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = ∫π‘Žπ‘Ž 𝑓𝑓(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 ± ∫π‘Žπ‘Ž 𝑔𝑔(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
𝑏𝑏
Integration by substitution:
𝑔𝑔(𝑏𝑏)
𝑔𝑔(𝑏𝑏)
∫π‘Žπ‘Ž 𝑓𝑓(𝑔𝑔(π‘₯π‘₯)) · 𝑔𝑔′(π‘₯π‘₯) 𝑑𝑑𝑑𝑑 = ∫𝑔𝑔(π‘Žπ‘Ž) 𝑓𝑓(𝑑𝑑) 𝑑𝑑𝑑𝑑 = [𝐹𝐹(𝑑𝑑)]𝑔𝑔(π‘Žπ‘Ž) = 𝐹𝐹�𝑔𝑔(𝑏𝑏)οΏ½ − 𝐹𝐹�𝑔𝑔(π‘Žπ‘Ž)οΏ½
b
∫ f ( x) ⋅ g ( x)dx = [ F ( x) ⋅ g ( x)]
Integration by parts:
b
a
a
−
b
∫ F ( x) ⋅ g´( x)dx
a
Volume of solid of revolution
b
Vx= π ⋅ ∫ f ( x) 2 dx
10
a
Differential Equations
Equation
Solution
𝑦𝑦 ′ = β„Ž(π‘₯π‘₯)
𝑦𝑦 = ∫ β„Ž(π‘₯π‘₯)dx
𝑦𝑦 ′ = 𝑏𝑏 − π‘Žπ‘Ž · 𝑦𝑦
𝑦𝑦 = + 𝑐𝑐 · 𝑒𝑒 −π‘Žπ‘Ž·π‘₯π‘₯
𝑦𝑦 ′ = π‘˜π‘˜ · 𝑦𝑦
𝑦𝑦 ′ = 𝑦𝑦 · (𝑏𝑏 − π‘Žπ‘Ž · 𝑦𝑦)
𝑦𝑦 ′ = π‘Žπ‘Ž · 𝑦𝑦 · (𝑀𝑀 − 𝑦𝑦)
𝑦𝑦 ′ + 𝑔𝑔(π‘₯π‘₯) · 𝑦𝑦 = β„Ž(π‘₯π‘₯)
Separation of variables:
( y´ =
dy
dx
x : the independent variable)
𝑦𝑦 = 𝑐𝑐 · 𝑒𝑒 π‘˜π‘˜·π‘₯π‘₯
𝑏𝑏
π‘Žπ‘Ž
𝑦𝑦 =
𝑦𝑦 =
𝑏𝑏
π‘Žπ‘Ž
1+𝑐𝑐·π‘’𝑒 −𝑏𝑏·π‘₯π‘₯
𝑀𝑀
1+𝑐𝑐·π‘’𝑒 −π‘Žπ‘Ž·π‘€π‘€·π‘₯π‘₯
𝑦𝑦 = 𝑒𝑒 −𝐺𝐺(π‘₯π‘₯) ∫ β„Ž(π‘₯π‘₯) · 𝑒𝑒 𝐺𝐺(π‘₯π‘₯) 𝑑𝑑𝑑𝑑
G ( x) : an antiderivative of g ( x)
y´=
h( x ) ⋅ g ( y ) ⇔
1
∫ g ( y) dy =∫ h( x) dx
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Geometry
Equiangular Triangles
π‘Žπ‘Ž1 𝑏𝑏1 𝑐𝑐1
= = = π‘˜π‘˜
π‘Žπ‘Ž
𝑏𝑏
𝑐𝑐
Right-angled Triangle
B
c
a
11
A
b
C
Pythagoras
cosine, sine & tangent in right-angled triangles
π‘Žπ‘Ž2 + 𝑏𝑏 2 = 𝑐𝑐 2
cos(𝐴𝐴) =
sin(𝐴𝐴) =
tan(𝐴𝐴) =
𝑏𝑏
𝑐𝑐
π‘Žπ‘Ž
𝑐𝑐
π‘Žπ‘Ž
𝑏𝑏
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Any triangle
B
c
a
A
b
C
Sine rule
𝑏𝑏
𝑐𝑐
π‘Žπ‘Ž
=
=
sin (𝐴𝐴) sin (𝐡𝐡) sin(𝐢𝐢)
Cosine rule
𝑐𝑐 2 = π‘Žπ‘Ž2 + 𝑏𝑏 2 − 2 · π‘Žπ‘Ž · 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐(𝐢𝐢)
𝐢𝐢 = cos−1 οΏ½
Area of triangle
𝑇𝑇 =
Circle
Area of circle
π‘Žπ‘Ž2 + 𝑏𝑏 2 − 𝑐𝑐 2
οΏ½
2π‘Žπ‘Žπ‘Žπ‘Ž
1
· π‘Žπ‘Ž · 𝑏𝑏 · 𝑠𝑠𝑠𝑠𝑠𝑠(𝐢𝐢)
2
𝐴𝐴 = πœ‹πœ‹ · π‘Ÿπ‘Ÿ 2
Circumference of circle
𝑂𝑂 = 2 · πœ‹πœ‹ · π‘Ÿπ‘Ÿ
Geometry in 2 D
Distance between 2 points 𝐴𝐴(π‘₯π‘₯1 , 𝑦𝑦1 ) og 𝐡𝐡(π‘₯π‘₯2 , 𝑦𝑦2 )
AB = οΏ½(π‘₯π‘₯1 − π‘₯π‘₯2 )2 + (𝑦𝑦1 − 𝑦𝑦2 )2
The midpoint M between two points 𝐴𝐴(π‘₯π‘₯1 , 𝑦𝑦1 ) og 𝐡𝐡(π‘₯π‘₯2 , 𝑦𝑦2 )
π‘₯π‘₯1 +π‘₯π‘₯2
2
M =οΏ½
,
𝑦𝑦1 +𝑦𝑦2
οΏ½
2
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12
Vectors in 2 D
→
Length of a vector π‘Žπ‘Žβƒ—
|π‘Žπ‘Žβƒ—| = οΏ½π‘Žπ‘Ž12 + π‘Žπ‘Ž22
a 
a =  1 
ο£­ a2 ο£Έ
Coordinates of a vector
π‘Žπ‘Ž1
π‘˜π‘˜ · π‘Žπ‘Ž1
π‘˜π‘˜ οΏ½π‘Žπ‘Ž οΏ½ = οΏ½
οΏ½
π‘˜π‘˜ · π‘Žπ‘Ž2
2
Multiplication by a scalar
∧
→
∧
→
→
Unit vector a in the direction of a :
a=
→
a
→
(the concept ”tværvektor” is not used outside DK)
a
Sum of vectors / difference of vectors
π‘Žπ‘Ž1
𝑏𝑏
π‘Žπ‘Ž ± 𝑏𝑏1
π‘Žπ‘Žβƒ— ± 𝑏𝑏�⃗ = οΏ½π‘Žπ‘Ž οΏ½ ± οΏ½ 1 οΏ½ = οΏ½ 1
οΏ½
𝑏𝑏2
π‘Žπ‘Ž2 ± 𝑏𝑏2
2
Dot product
𝑏𝑏�⃗
π‘Žπ‘Žβƒ— · 𝑏𝑏�⃗ = π‘Žπ‘Ž1 𝑏𝑏1 + π‘Žπ‘Ž2 𝑏𝑏2
→
→
→ 2
a⋅a= a
v
π‘Žπ‘Žβƒ—
π‘Žπ‘Žβƒ— · 𝑏𝑏�⃗ = |π‘Žπ‘Žβƒ—||𝑏𝑏�⃗| · cos (𝑣𝑣)
v is the angle between π‘Žπ‘Žβƒ— and 𝑏𝑏�⃗.
Perpendicular vectors
π‘Žπ‘Žβƒ—
π‘Žπ‘Žβƒ— · 𝑏𝑏�⃗ = 0 ⇔ π‘Žπ‘Žβƒ— ⊥ 𝑏𝑏�⃗
𝑏𝑏�⃗
Projection of 𝑏𝑏�⃗ on π‘Žπ‘Žβƒ—
𝑏𝑏�⃗
π‘π‘οΏ½βƒ—π‘Žπ‘Ž
π‘Žπ‘Žβƒ—
π‘π‘οΏ½βƒ—π‘Žπ‘Ž =
Length of projection vector
π‘Žπ‘Žβƒ— · 𝑏𝑏�⃗
π‘Žπ‘Žβƒ—
|π‘Žπ‘Žβƒ—|2
|π‘π‘οΏ½βƒ—π‘Žπ‘Ž | =
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οΏ½π‘Žπ‘Žβƒ— · 𝑏𝑏�⃗�
|π‘Žπ‘Žβƒ—|
13
Determinant of vectors
detοΏ½π‘Žπ‘Žβƒ—, 𝑏𝑏�⃗� = π‘Žπ‘Ž1 𝑏𝑏2 − π‘Žπ‘Ž2 𝑏𝑏1 = οΏ½
π‘Žπ‘Ž1
π‘Žπ‘Ž2
detοΏ½π‘Žπ‘Žβƒ—, 𝑏𝑏�⃗� = |π‘Žπ‘Žβƒ—||𝑏𝑏�⃗| · sin(𝑣𝑣)
𝑏𝑏1
οΏ½
𝑏𝑏2
v is the angle from π‘Žπ‘Žβƒ— to 𝑏𝑏�⃗ (positive anticlockwise and
negative clockwise)
Parallel vectors:
detοΏ½π‘Žπ‘Žβƒ—, 𝑏𝑏�⃗� = 0 ⇔ π‘Žπ‘Žβƒ— βˆ₯ 𝑏𝑏�⃗
Area of the parallelogram spanned by π‘Žπ‘Žβƒ— og 𝑏𝑏�⃗
𝐴𝐴 = |det (π‘Žπ‘Žβƒ—, 𝑏𝑏�⃗)|
𝑏𝑏�⃗
π‘Žπ‘Žβƒ—
14
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Vectors in 3 D
Most definitions and theorems concerning vectors in 2 D can be transferred to 3 D by adding a 3rd
coordinate.
Specific definitions and theorems for vectors in 3 D:
Vektor product (cross product)
π‘Žπ‘Žβƒ— × π‘π‘οΏ½βƒ—
𝑏𝑏�⃗
v
π‘Žπ‘Ž
οΏ½ 2
π‘Žπ‘Ž
βŽ› 3
π‘Žπ‘Ž
π‘Žπ‘Žβƒ— × π‘π‘οΏ½βƒ— = ⎜� 3
⎜ π‘Žπ‘Ž1
π‘Žπ‘Ž
οΏ½ 1
⎝ π‘Žπ‘Ž2
Lenght of a vector product
𝑏𝑏2
οΏ½
𝑏𝑏3
⎞
𝑏𝑏3 ⎟
οΏ½
𝑏𝑏1 ⎟
𝑏𝑏1
οΏ½
𝑏𝑏2 ⎠
οΏ½π‘Žπ‘Žβƒ— × π‘π‘οΏ½βƒ—οΏ½ = |π‘Žπ‘Žβƒ—||𝑏𝑏�⃗|sin (𝑣𝑣)
π‘Žπ‘Žβƒ—
v is the angle between π‘Žπ‘Žβƒ— and 𝑏𝑏�⃗ .
π‘Žπ‘Žβƒ— × π‘π‘οΏ½βƒ—
Area of the parallelogram spanned by π‘Žπ‘Žβƒ— and 𝑏𝑏�⃗
𝐴𝐴 = οΏ½π‘Žπ‘Žβƒ— × π‘π‘οΏ½βƒ—οΏ½
𝑏𝑏�⃗
v
π‘Žπ‘Žβƒ—
Geometry in 3 D
Sphere
Equation of a sphere with centre 𝐢𝐢(π‘₯π‘₯0 , 𝑦𝑦0 , 𝑧𝑧0 ) and radius r
(π‘₯π‘₯ − π‘₯π‘₯0 )2 + (𝑦𝑦 − 𝑦𝑦0 )2 + (𝑧𝑧 − 𝑧𝑧0 )2 = π‘Ÿπ‘Ÿ 2
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15
Line in 3 D
Parametric form of the line l through 𝑃𝑃0 (π‘₯π‘₯0 , 𝑦𝑦0 , 𝑧𝑧0 ) with
π‘Ÿπ‘Ÿ1
direction vector π‘Ÿπ‘Ÿβƒ— = οΏ½π‘Ÿπ‘Ÿ2 οΏ½
π‘Ÿπ‘Ÿ3
π‘₯π‘₯0
π‘Ÿπ‘Ÿ1
π‘₯π‘₯
�𝑦𝑦� = �𝑦𝑦0 οΏ½ + 𝑑𝑑 οΏ½π‘Ÿπ‘Ÿ2 οΏ½
𝑧𝑧0
π‘Ÿπ‘Ÿ3
𝑧𝑧
Distance between the point P and the line l intersecting
the point 𝑃𝑃0 and having the direction vector π‘Ÿπ‘Ÿβƒ—
P
π‘Ÿπ‘Ÿβƒ—
𝑃𝑃0
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(𝑃𝑃, 𝑙𝑙) =
l
|π‘Ÿπ‘Ÿβƒ— × οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½βƒ—
𝑃𝑃0 𝑃𝑃|
|π‘Ÿπ‘Ÿβƒ—|
16
Plane in 3 D
𝑛𝑛�⃗
α
P0
The equation of a plane α through 𝑃𝑃0 (π‘₯π‘₯0 , 𝑦𝑦0 , 𝑧𝑧0 ) where
π‘Žπ‘Ž
𝑛𝑛�⃗ = �𝑏𝑏� is a vector perpendicular to the plane
𝑐𝑐
π‘Žπ‘Ž · (π‘₯π‘₯ − π‘₯π‘₯0 ) + 𝑏𝑏 · (𝑦𝑦 − 𝑦𝑦0 ) + 𝑐𝑐 · (𝑧𝑧 − 𝑧𝑧0 ) = 0
P
α
Distance between the point 𝑃𝑃(π‘₯π‘₯1 , 𝑦𝑦1 , 𝑧𝑧1 ) and the plane
α given by π‘Žπ‘Ž · π‘₯π‘₯ + 𝑏𝑏 · 𝑦𝑦 + 𝑐𝑐 · 𝑧𝑧 + 𝑑𝑑 = 0
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑(𝑃𝑃, 𝛼𝛼) =
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|π‘Žπ‘Ž · π‘₯π‘₯1 + 𝑏𝑏 · 𝑦𝑦1 + 𝑐𝑐 · 𝑧𝑧1 + 𝑑𝑑|
√π‘Žπ‘Ž2 + 𝑏𝑏 2 + 𝑐𝑐 2
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