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Level 537

Vectors, Parametric & Polar Equations, the Complex


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Magnitude
The number of degrees a regular polygon can be rotated to map onto itself.
Zero vector
<0, 0>, denoted 0
Sum of vectors
u + v = <u₁, u₂> + <v₁, v₂> = <u₁ + v₁, u₂ + v₂>
Product of a vector and a scalar
k×u = k<u₁, u₂> = <k×u₁, k×u₂>
Unit Vector
a vector with a magnitude of 1. the positive X-axis is vector i, pos. <1,0> y xis is vector j <0,1>
i
<1, 0 >
j
<0, 1>
Solving for a*i* + b*j* form
v = (|v| cosθ)i + (|v| sinθ)j
Dot product
x times x, y times y, then add them together
Angle between two vectors
cosθ = (u • v)/(|u|×|v|)
Orthogonal vectors
Condition that is true if and only if u • v = 0
Projection of u onto v
proj,v u = v × ((u • v)/(|v|²))
Parametric equation
Equation of a function defined on an interval I of t-values, in the form x = f(t) and y = g(t)
Ferris wheel type problems
radius r, center (a, b), spinning at k rad/sec,
Throwing a ball at a Ferris wheel problems
Radius r, center (a, b), spinning at k rad/sec,
Polar coordinates
Pol(x, y) = (√(x² + y²), tan⁻¹(y/x))
Rectangular coordinates
Rect(r, θ) = (r cosθ, r sinθ)
Distance with polar coordinates
Law of cosines, d = √(r₁² + r₂² - 2r₁r₂cos(θ₁-θ₂))
Limaçon curves
Graphs in the form r = a ± b sinθ and r = a ± b cosθ
Rose curves
Graphs in the form r = a cos nθ and r = a sin nθ, where n > 1, and if n is odd, there are n petals, and if even, there are 2n petals
Lemniscate curves
Graphs in the form r² = a² sin 2θ and r² = a² cos 2θ
complex number
includes real and imaginary numbers
r(cosθ + i sinθ)
Trigonometric form of a complex number
Multiplication using trigonometric complex numbers
z₁×z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
Division using trigonometric complex numbers
z₁/z₂ = (r₁/r₂)[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)], r₂≠0
De Moivre's Theorem
zⁿ = rⁿ(cos nθ + i sin nθ)
Roots of a complex number
the nth root of [r×[cos((θ+2πk)/n) + i sin((θ+2πk)/n)]], where k = 0, 1, ... n - 1