Mathematics reference
Rules for vectors


Basic properties of vectors.


Legend.

 u, v, and w represent vectors,
 o is the zero vector,
 r and s represent scalars,
 t represents the independent variable,
 theta represents the angle between u and v,
 d is the differential operator,
 dot represents the dot product,
 cross represents the cross product,
 ... represents the norm.

Basic.


u == (1) u

equation 1


u  v == u + (v)

equation 2


1 v = v

equation 3


0 v = o

equation 4


v + o = v

equation 5


v  v = o

equation 6


unit v = (1/v) v, v != o

equation 7


Addition and scalar product.


u + v = v + u

equation 8


(u + v) + w = u + (v + w)

equation 9


r (u + v) = r u + r v

equation 10


(r + s) u = r u + s u

equation 11


(r s) v = r (s v)

equation 12


Vector products.


u dot v = v dot u

equation 13


(u + v) dot w = u dot w + v dot w

equation 14


u dot (v + w) = u dot v + u dot w

equation 15


(r u) dot v = u dot (r v) = r (u dot v)

equation 16


v dot o = o dot v = 0

equation 17


v dot v = v^{2}

equation 18


u dot v = u v cos theta

equation 19


u cross v = v cross u

equation 20


(u + v) cross w = u cross w + v cross w

equation 21


u cross (v + w) = u cross v + u cross w

equation 22


(r u) cross v = u cross (r v) = r (u cross v)

equation 23


v cross o = o cross v = o

equation 24


v cross v = o

equation 25


u cross v = u v sin theta

equation 26


u dot v cross w = u cross v dot w

equation 27


Derivatives.


(d/dt) (u + v) = du/dt + dv/dt

equation 28


(d/dt) (r u) = dr/dt v + r du/dt

equation 29


(d/dt) (u dot v) = du/dt dot v + u dot dv/dt

equation 30


(d/dt) (u cross v) = du/dt cross v + u cross dv/dt

equation 31


Unit basis vectors in R^{3}.


i == (1, 0, 0)

equation 32


j == (0, 1, 0)

equation 33


k == (0, 0, 1)

equation 34


i dot i = j dot j = k dot k = 1

equation 35


i dot j = j dot k = k dot i = 0

equation 36


i cross j = k

equation 37


j cross k = i

equation 38


k cross i = j

equation 39


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