Core Pure Flashcards
What is |z1z2| equivalent to?
|z1||z2|
(Same logic for division)
What is arg(z1z2) equivalent to?
arg(z1) + arg(z2)
(Same logic for division)
What can the determinant of a matrix tell you?
The determinant of a matrix is the scale factor for the area or volume of shapes before and after the transformation.
What 2x2 matrix rotates a point anticlockwise by θº about the origin?
[cosθ -sinθ]
[sinθ cosθ]
What 3x3 matrix rotates a point θº anticlockwise around the positive x axis?
[1 0 0]
[0 c -s]
[0 s c]
What 3x3 matrix rotates a point θº anticlockwise around the positive y axis?
[c 0 s]
[0 1 0]
[-s 0 c]
What 3x3 matrix rotates a point θº anticlockwise around the positive z axis?
[c -s 0]
[s c 0]
[0 0 1]
How can you convert from vector line equation to cartesian line equation?
r = (x1, y1, z1) + λ(a, b, c)
(x - x1)/a = (y - y1)/b = (z - z1)/c
How can you simplify arctan(a) + arctan (b)?
arctan(a) + arctan (b) = arctan( (a + b) / (1- ab) )
What is simple harmonic motion?
When the acceleration of a particle is always towards a fixed point, on a line of motion, where the acceleration is proportional to the displacement of the particle.
What is the equation for simple harmonic motion?
ẍ = -ω²x
- ω is the angular velocity of the particle.
- ω is the coefficient of t in the sin and cos function of the solved equation
- The minus sign meaning the acceleration is towards the origin.
What are the different types of simple harmonic motion?
ẍ + kẋ + ω²x = f(t)
f(t) = 0:
- Light dampening (k² < 4ω²): a force is reducing the amplitude of the oscillations
- Heavy dampening (k² > 4ω²): a force is resisting the motion so much that no oscillations are experienced
- Critical dampening (k² = 4ω²): the minimum resistive force that results in heavy dampening
f(t) ≠ 0
- Forced harmonic motion: other than the other two forces, the particle experiences another external force
