Vectors

Fig.1. Vectors a and b, a - b, and representing them as sum of orthogonal vectors.

If we have vectors a = OA and b = OB, then b - a = AB.

A projection of the vector b onto direction of the vector a can be calculated using scalar vector product (a.k.a. dot-product) as follows.

(1) OC = DB = a ·(b·a)/(a·a)

The other component of the vecor b in an orthogonal cartesian coordinates can be calculated using vector cross-product and dot-product, or only dot-product.

(2) CB = OD = [[a×b]×a]/(a·a) = [a×[b×a]]/(a·a) = b - a·(a·b)/(a·a)

The vector a can be presented in a similar way, as shown in Fig.1.

Additionally, the vector [a×b] is perpendicular to the plane of vectors a and b and is forwarded towards the viewer, while [b×a] has the opposite direction. The length of these vectors is area of a parallelogram defined by a and b.

(3) a·b = b·a
(4) [a×b] = -[b×a]

If α is the angle beween a and b, then

(5) |[a×b]| = |[b×a]| = |a| ·|b| sin α
(6) a·b = b·a = |a| ·|b| cos α
(7) [[a×b]×c] = [c×[b×a]] = -[[b×a]×c] = -[c×[a×b]]

Also, the mixed product (or the scalar triple product), a⋅(b×c), is the (signed) volume of the parallelepiped defined by the three vectors given. It is invariant under circular shift of the three parameters and changes sign under swapping of any two parameters.

(8) a⋅[b×c]=[a×b]⋅c=[b×c]⋅a=b⋅[c×a]=[c×a]⋅b=c⋅[a×b]=-[a×c]⋅b=-a⋅[c×b]=-[b×a]⋅c=-b⋅[a×c]=-[c×b]⋅a=-c⋅[b×a]

The rule "bac minus cab" allows to calculate triple cross-product using dot-products.

(9) [a×[b×c]] = b· (a⋅c) - c· (a⋅b)

Occasionally, it's written differently, contrary to the above mnemonics (Fig.1 and formula (2) present a special case of this form of the rule):

(10) [[a×b]×c] = b ·(a⋅c) - a ·(b⋅c)

Other useful equations:
(11) [a×b]⋅[c×d] = (a⋅c)⋅(b⋅d) - (a⋅d)⋅(b⋅c)
(12) [a×b]×[c×d] = (a⋅[b×d])⋅c - (a⋅[b×c])⋅d = (a⋅[c×d])⋅b - (b⋅[c×d])⋅a
(13) [a×b]×[a×c] = (a⋅[b×c])⋅a

Components over a general linear coordinate basis a, b, c can be determined according to the equation
(14) d = ((d⋅[b×c])⋅a + (a⋅[d×c])⋅b + (a⋅[b×d])⋅c)/(a⋅[b×c])
where a⋅[b×c]!=0

(15) [a×b]×c + [b×c]×a + [c×a]×b = o
(16) [a×b]×c = a×[b×c] + b×[c×a]