Mathematics reference
Rules for vectors
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Basic properties of vectors.
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Legend.
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- 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.
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Basic.
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-u == (-1) u
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equation 1
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u - v == u + (-v)
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equation 2
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1 v = v
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equation 3
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0 v = o
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equation 4
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v + o = v
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equation 5
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v - v = o
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equation 6
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unit v = (1/|v|) v, v != o
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equation 7
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Addition and scalar product.
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u + v = v + u
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equation 8
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(u + v) + w = u + (v + w)
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equation 9
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r (u + v) = r u + r v
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equation 10
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(r + s) u = r u + s u
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equation 11
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(r s) v = r (s v)
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equation 12
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Vector products.
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u dot v = v dot u
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equation 13
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(u + v) dot w = u dot w + v dot w
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equation 14
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u dot (v + w) = u dot v + u dot w
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equation 15
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(r u) dot v = u dot (r v) = r (u dot v)
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equation 16
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v dot o = o dot v = 0
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equation 17
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v dot v = |v|2
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equation 18
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u dot v = |u| |v| cos theta
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equation 19
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u cross v = -v cross u
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equation 20
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(u + v) cross w = u cross w + v cross w
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equation 21
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u cross (v + w) = u cross v + u cross w
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equation 22
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(r u) cross v = u cross (r v) = r (u cross v)
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equation 23
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v cross o = o cross v = o
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equation 24
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v cross v = o
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equation 25
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|u cross v| = |u| |v| sin theta
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equation 26
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u dot v cross w = u cross v dot w
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equation 27
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Derivatives.
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(d/dt) (u + v) = du/dt + dv/dt
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equation 28
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(d/dt) (r u) = dr/dt v + r du/dt
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equation 29
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(d/dt) (u dot v) = du/dt dot v + u dot dv/dt
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equation 30
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(d/dt) (u cross v) = du/dt cross v + u cross dv/dt
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equation 31
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Unit basis vectors in R3.
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i == (1, 0, 0)
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equation 32
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j == (0, 1, 0)
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equation 33
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k == (0, 0, 1)
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equation 34
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i dot i = j dot j = k dot k = 1
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equation 35
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i dot j = j dot k = k dot i = 0
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equation 36
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i cross j = k
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equation 37
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j cross k = i
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equation 38
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k cross i = j
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equation 39
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Navigation.
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Mathematics reference: Rules for integration
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Basic properties of matrices.
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