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Hint

1

3x3 matrix for a rotation of θ degrees anticlockwise about the y axis

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2

Rotation of θ degrees anticlockwise about the x axis

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3

Rotation of θ degrees anticlockwise about the z axis

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4

α + β

-b / a

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5

αβ

c / a

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6

α + β + γ

-b / a

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7

αβγ

-d / a

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8

αβ + γβ + αγ

c / a

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9

Σα

α + β + γ

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10

Σαβ

αβ + γβ + αγ

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11

3x3 matrix of a reflection in x = 0 (the y-z plane)

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12

3x3 matrix of a reflection in y = 0 (the x-z plane)

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13

3x3 matrix of a reflection in z = 0 (the x-y plane)

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14

α + β + γ + σ

-b / a

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15

αβγσ

e / a

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16

αβ + αγ + ασ +βγ + βσ + γσ

c / a

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17

αβγ + αβσ + αγσ + βασ

-d / a

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18

The formula for the perpendicular distance from the point (α,β,γ) to a plane

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19

if f(x) = c, what would the substitution for the particular integral be?

a

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20

if f(x) = bx+c, what would the substitution for the particular integral be?

ax + b

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21

if f(x) = ax² + bx +c , what would the substitution for the particular integral be?

ax² + bc + c

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22

if f(x) = Pe^(px), what would the substitution for the particular integral be?

Qe^(px)

OR Qxe^(px)

OR Qx²e^(px)

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23

if f(x) = rcos(wx) + s sin(wx), what would the substitution for the particular integral be?

pcos(wx) + qsin(wx)

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24

If an object is moving with simple harmonic motion…

the only force on it is proportional to the displacement, but in the opposite direction

i.e. F=-kx

the force increases as you get further from the starting position

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25

So F=ma…

-kx = m a = mx:

which means x: = -k/m (x)

WHICH IS OFTEN WRITTEN AS x: = -w²x

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26

When is simple harmonic motion simple?

when it assumes there’s no resistive force (e.g. friction)

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27

What is damping?

the act of stopping the oscillations

then more realistic

resistance can be modelled as being proportional to the velocity, acting in the direction opposite to motion

i.e. -r dx/dt

THUS HAS THE 3 TYPES OF DAMPING (underdamped, critically damped and overdamped)

m² + rm + x = 0

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28

What’s the difference between free vibrations and forced vibrations?

Free - they’re set in motion and then just left to it

m² + rm + x = 0

Forced - if you add an external force

m² + rm + x = f(t)

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29

What is the general solution to x: = -w²x ?

x = Acoswt + Bsinwt

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30

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