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
How can you use complex numbers to solve this problem?
A particle moves one unit then turns an angle of α to the right. It does this four times in total. What is the distance from original to final position?
- Original position is origin.
- First moves by 1: distance is 1
- Then moves by complex number e^αi
- Then moves by complex number e^2αi, and so on
- Sum these numbers to find the final displacement
- Calculate this sum by doing a sum of a geometric series.
How do you solve an equation in the form:
cosnθ = …
sinnθ = …
- let z = cosθ + isinθ
- zⁿ = cosnθ + isinnθ
- cosnθ = Re(zⁿ) = Re((cosθ + isinθ)ⁿ)
- sinnθ = Im(…)
- Expand binomially
- Use sin²θ + cos²θ = 1 where appropriate
Remember: If finding real components, write imaginary terms of the expansion as +i(…), and if finding imaginary, write real as +(…) to save time.
How do you solve an equation in the form:
cosⁿθ = …
sinⁿθ = …
E.g. for cosⁿθ
- let z = cosθ + isinθ
- z + z⁻¹ = 2cosθ
- cosθ = (1/2)(z + z⁻¹)
- cosⁿθ = (1/2)(z + z⁻¹)ⁿ
- Expand binomially
- Collect terms using de Moivre’s theorem
What’s the formula for arc length, sector area and segment area?
s = rθ
Asec = (1/2)r²θ
Aseg = (1/2)r²(θ - sinθ)
How can you quickly multiply a matrix by a vector where the matrix is in terms of linear k?
Result =
(Matrix of constants)(Vector) + k(Matrix of coefficients)(Vector)
What must you remember when finding the area between two polar curves?
You MUST work out the different areas separately and subtract them. Subtracting the polar equations before integrating is incorrect because the radius is scaled down.
When finding the intersection between a plane and a line, what method should you use?
If line is in vector form and plane is in cartesian, substitute x, y and z into the cartesian in terms of λ from the vector equation.
Solve for λ and find coordinate.
What are tips for tricky integrations?
- Consider working both ways with the double angle formulae: a double angle can reduce the power of a trig expression, or a single angle could allow for a reverse chain rule to be used.
- For modelling questions with decimals, keep in terms of decimals (e.g. keep 1/3.6² factorised out or whatever) so that numbers are as simple to work with as possible. Decimals can be less clunky than fractions.
What equations relate cartesian to polar coordinates?
x = rcosθ
y = rsinθ
What should you include when sketching the positions of complex numbers on an Argand diagram?
Just mark them as dots or xs and label the axis that they are in line with. If defined by mod arg, draw angle and line.
How do you know if a polar curve has a dimple or not?
Put in form r = a(b+cosθ)
If 1<|b|<2, there is a dimple.
Similar for sinθ
How can you transform a shape in the complex plane so that its around the centre and has a vertex of 1?
- Take a point z1 on the shape.
- Find α, the centre.
- z1 - α centres the coordinate.
- β(z1 - α) rotates and scales the coordinate so it lands on 1.
- β = 1 / (z1 - α)
How can you simplify trig functions of inverse trig functions, e.g. sin(4arccos 1/√3)
if x = arccos 1/√3, cos x = 1/√3
- We want an expression in terms of cos x
- Take sin4x and use the double angle formulae
- use sin x = √( 1 - cos²x) for any remaining sin x
How would you go about finding the real and imaginary components of 1 / (1 - 2e^θi)
Multiply both sides of the fraction by (1 - 2e^-θi)
What is the strategy for converting polar to cartesian?
- If required, use double angle formula to get all trig functions in terms of θ.
- Keep multiplying and dividing by r to make all trig functions either rcosθ or rsinθ, then replace them with x and y.
- Keep multiplying and dividing by r to make each r term r², then replace with x² + y².
- Simplify.
