Equations Flashcards
Formula for P distance from Origin
r^2 = x^2 + y^2 + z^2
Angle OP makes with each axis
cos(thetre) = (x,y or z)/r(length of OP)
3D axis symbols
z axis - gamma
y axis - beta
x axis - alpha
angle relationship
cos^2 (alpha) + cos^2 (beta) + cos^2 (gamma) = 1
Finding the unit vector
divide x, y and z by each respective modulus.
Vector addition
parallelogram rule
A + B = (a1 + b1, a2 + b2, a3 + b3)
Associative Law for addition of vectors
(a + b) + c = a + (b + c)
It doesn’t matter which order you add you still get the same vector.
Distributive law
(lander)(a + b) = (lander)a + (lander)b
How to convert a vector with magnitude and direction into cartesian form
Find unit vector in direction then multiply by magnitude
Work done equation
((mag of) a)((mag of) F)(cos(thetre))
a.b
a1b1 + a2b2 + a3b3
cross product
a x b = |a| |b| sin(thetre) â
where a and b are vectors and
where â is normal to vectors a and b
cross product area of a parallelogram
A = a x b
where a and b are vectors
cross product area of triangle
A = 0.5(a x b)
where a and b are vectors
unit vectors
i . i
1
unit vectors
i . j
0
unit vectors
i x i
0
unit vectors
i x j
k
cross product in component form
(a2b3-a3b2)i - (a1b3-a3b1)j + (a1b2-a2b1)k
orthogonality with cross product unit vectors
clockwise is positive anti clockwise is negative.
eg
i x j = k, j x k = i, k x i = j
where as
j x i = -k, k x j = -i, i x k = -j
cross product distributive law
a x (b+c) = (a x c) + (b x c)
Principle of transmissibility
when finding moment of a force
M = r1 x F = r2 x F =… = rn x F
therefore we find the easiest value of r to use, often one of the axis in the plane the force intercepts
component of force in another vector
(a.b)/(mag of b)
vector triple product
(a x b) x c = b(a.c) - a(b.c)
