Mathematics reference
Rules for vectors
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MathRef
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 R3.
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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