Vector algebra relations

Magnitudes

The magnitude of a vector A is determined by its three components along three orthogonal directions using Pythagoras' theorem:

∥ A ∥ 2 = A 1 2 + A 2 2 + A 3 2  

The magnitude also can be expressed using the dot product:

∥ A ∥ 2 = ( A ⋅ A )  

Inequalities

A ⋅ B ∥ A ∥ ∥ B ∥ ≤ 1   ; Cauchy–Schwarz inequality in three dimensions ∥ A + B ∥ ≤ ∥ A ∥ + ∥ B ∥ ; the triangle inequality in three dimensions ∥ A − B ∥ ≥ ∥ A ∥ − ∥ B ∥ ; the reverse triangle inequality

Here the notation (A · B) denotes the dot product of vectors A and B.

Angles

The vector product and the scalar product of two vectors define the angle between them, say θ:

sin ⁡ θ = ∥ A × B ∥ ∥ A ∥ ∥ B ∥     ( − π < θ ≤ π )

To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise.

cos ⁡ θ = A ⋅ B ∥ A ∥ ∥ B ∥     ( − π < θ ≤ π )

Here the notation A × B denotes the vector cross product of vectors A and B. The Pythagorean trigonometric identity then provides:

∥ A × B ∥ 2 + ( A ⋅ B ) 2 = ∥ A ∥ 2 ∥ B ∥ 2

If a vector A = (A_x, A_y, A_z) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then:

cos ⁡ α = A x A x 2 + A y 2 + A z 2 = A x ∥ A ∥   ,

and analogously for angles β, γ. Consequently:

A = ∥ A ∥ ( cos ⁡ α   i ^ + cos ⁡ β   j ^ + cos ⁡ γ   k ^ )   ,

with i ^ ,   j ^ ,   k ^ unit vectors along the axis directions.

Areas and volumes

The area Σ of a parallelogram with sides A and B containing the angle θ is:

Σ = A B   sin ⁡ θ   ,

which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is:

Σ = ∥ A × B ∥ = ∥ A ∥ 2 ∥ B ∥ 2 − ( A ⋅ B ) 2   .

The square of this expression is:

Σ 2 = ( A ⋅ A ) ( B ⋅ B ) − ( A ⋅ B ) ( B ⋅ A ) = Γ ( A ,   B )   ,

where Γ(A, B) is the Gram determinant of A and B defined by:

Γ ( A ,   B ) = | A ⋅ A A ⋅ B B ⋅ A B ⋅ B |   .

In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A, B and C is given by the Gram determinant of the three vectors:

V 2 = Γ ( A ,   B ,   C ) = | A ⋅ A A ⋅ B A ⋅ C B ⋅ A B ⋅ B B ⋅ C C ⋅ A C ⋅ B C ⋅ C |   .

This process can be extended to n-dimensions.

Addition and multiplication of vectors

Some of the following algebraic relations refer to the dot product and the cross product of vectors. These relations can be found in a variety of sources, for example, see Albright.

c ( A + B ) = c A + c B ; distributivity of multiplication by a scalar and addition

A + B = B + A ; commutativity of addition

A + ( B + C ) = ( A + B ) + C ; associativity of addition

A ⋅ B = B ⋅ A ; commutativity of scalar (dot) product

A × B = − B × A ; anticommutativity of vector cross product

( A + B ) ⋅ C = A ⋅ C + B ⋅ C ; distributivity of addition wrt scalar product

( A + B ) × C = A × C + B × C ; distributivity of addition wrt vector cross product

A ⋅ ( B × C ) = B ⋅ ( C × A ) = ( A × B ) ⋅ C

A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B ) C ; vector triple product

( A × B ) ⋅ ( C × D ) = ( A ⋅ C ) ( B ⋅ D ) − ( B ⋅ C ) ( A ⋅ D ) ; Binet–Cauchy identity in three dimensions

In particular, when A = C and B = D, the above reduces to: ( A × B ) ⋅ ( A × B ) = | A × B | 2 = ( A ⋅ A ) ( B ⋅ B ) − ( A ⋅ B ) 2 ; Lagrange's identity in three dimensions

[ A , B , C ] D = ( A ⋅ D ) ( B × C ) + ( B ⋅ D ) ( C × A ) + ( C ⋅ D ) ( A × B )

A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:

( A × B ) × ( C × D ) = [ A , B , D ] C − [ A , B , C ] D = [ A , C , D ] B − [ B , C , D ] A where [A, B, C] is the scalar triple product A · (B × C) or the determinant of the matrix {A, B, C} with the components of these vectors as columns .

Given three arbitrary vectors not on the same line, A, B, C, any other vector D can be expressed in terms of these as:

D = D ⋅ ( B × C ) [ A ,   B ,   C ]   A + D ⋅ ( C × A ) [ A ,   B ,   C ]   B + D ⋅ ( A × B ) [ A ,   B ,   C ]   C   .