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Level 359

Vectors & Complex Numbers


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vector (definition)
Has both magnitude and direction
Course
Intended path (With angle and direction) of a vehicle
Drift angle
Degree of drift due to conditions (wind, etc.)
Ax/A = cosθx
Solving for angles of 3D vectors <Ax, Ay, Az>
Unit Vectors
Unit vector ú (assuming A is the original) =
i, j, k notation
(X-component)·i + (y-component)·j + (z-component)·k ***Note-All have vector signs above
Dot products
a·b = |a|·|b|·cosθ (a and b are vectors)
Parallels
U|| = (U·V)/|U| (U and V are vectors) || == parallel
Projections
Proj(v)U = ((U·V)/|V|^2)·V (read Projection of U onto v and U and V are vectors)
Perpendiculars
U⊥=U-(Proj(v)U)
U||V if Ux/Vx=Uy/Vy=Uz/Vz
How to determine whether two vectors are either parallel or anti-parallel
Center of mass
Mass1/Mass2 = Dist1/Dist2
Collinear points
Points P, Q, and R are collinear if:
m·n matrix
m--> number of rows
matrix algebra
A+B = |a11+b11, a21+b21|
x-products
UxV = (UyVz-UzVy)i-(UxVz-UzVx)j+(UxVy-UyVx)k
Finding area of a parallelogram using x-product
Area of parallelogram = magnitude of x-product of the two vectors
Triple scalar
U·(VxW) = Volume of parallelepiped (U, V, and W are vectors)
z=a+bi
Rectangular form of complex numbers
Complex conjugates
Two complex numbers of the form (a+bi) and (a-bi).
z=r(cosθ+isinθ)
Trig form of complex numbers
z·x= r(z)·r(w)·(cos(θ(z)+θ(w))+isin(θ(z)+θ(w))
Multiplying and dividing complex numbers in trig form
r(cosθ+isinθ)^n=r^n(cos(nθ)+isin(nθ))
DeMoiuve's Theorem
Odd and even functions
even--> f(-x) = f(x) (cos is even)