MATH1131 Mathematics 1A (Algebra) Cheatsheet PDF

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MATH1131

2 Vectors

Polar Form

Mathematics 1A

z = rcisθ = r(cos θ + i sin θ)

University of New South Wales

Where r is the modulus ||z|| and θ the argument arg z .

1 Complex Numbers

Polar Form Operations and Properties

|v| =

Where:

i2 = −1

Modulus |z| =

p

a2 + b2

z = a + ib then

z = z − ib

 ˆ i v × w =  vx wx

Euler’s Formula

vw =

 kˆ  vz  wz 

v·w |w|

Vector projection v in direction of w

De Moivre’s Theorem Given z = reiθ :

vw = z n = rn einθ

Cos and Sin in Terms of Exponentials cos x =

ˆj vy wy

Scalar projection v in direction of w

Where e is Euler’s Number and x is real.

Properties of Modulus and Conjugate |zw| = |z||w|   |z| z   w |w|

Cross Product

arg(zw) = arg(z) + arg(w) z = arg(z) − arg(w) arg w

eix = cos x + i sin x

If

x2 + y 2 + z 2

v · w = vx wx + vy wy + vz wz = |v||w|cosθ

r rcis(θ) = cis(θ − φ) pcis(φ) p

Conjugate

p

Dot Product

rcis(θ) × pcis(φ) = rpcis(θ + φ) z = a + ib

Length of a vector

eix + e−ix 2

sin x =

eix − e−ix 2i



v·w |w|2

[u, v, w] = u · (v × w) v×w

V = [u, v, w] u w

z±w =z±w

v

zw = zw

Volume of a parallelpiped 1

w

Scalar Triple

|z + w| ≤ |z| + |w|

|z − w| ≥ |z| − |w|



3 Lines in 3D

5 Matrices

Parametric Equation of a Line

Matrix Multiplication

  x(t) = a + pt y(t) = b + qt   z(t) = c + rt

AB =

=

Symmetric Form of Line Equation x−a z−c y−b = = r q p

Vector Equation of a Line

Cartesian Equation of a Plane ax + by + cz = d

a b x y

aα + bβ + cγ xα + yβ + zγ

aρ + bσ + cτ xρ + yσ + zτ



Determinants 2x2 Matrix det

r(t) = d + tv

4 Planes in 3D



   α ρ c  β σ z γ τ



3x3 Matrix  a  d  g

b e h

   a b  a b  = ad − cb =  c d c d



    c   e f  − b d f  = a  g  h i  i

Parametric Equation of a Plane    x(u, v) = a + pu + lv  y(u, v) = b + qu + mv  z(u, v) = c + ru + nv 

Vector Equation of a Plane n · (r − d) = 0

2

  d f  + c   g i

 e   h...