P-adic exponential function

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Magnitudes

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

$\Vert \mathbf{A}{\Vert }^2=A_1^2+A_2^2+A_3^2$

The magnitude also can be expressed using the dot product:

$\Vert \mathbf{A}{\Vert }^2=(\mathbf{A}\mathbf{\cdot }\mathbf{A})$

Inequalities

$\frac{\mathbf{A}\mathbf{\cdot }\mathbf{B}}{\Vert \mathbf{A}\Vert \Vert \mathbf{B}\Vert }\le 1$; Cauchy–Schwarz inequality in three dimensions

$\Vert \mathbf{A}+\mathbf{B}\Vert \le \Vert \mathbf{A}\Vert +\Vert \mathbf{B}\Vert$; the triangle inequality in three dimensions

$\Vert \mathbf{A}\mathbf{-}\mathbf{B}\Vert \ge \Vert \mathbf{A}\Vert -\Vert \mathbf{B}\Vert$; 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:

$\sin \theta =\frac{\Vert \mathbf{A}\mathbf{\times }\mathbf{B}\Vert }{\Vert \mathbf{A}\Vert \Vert \mathbf{B}\Vert }(-\pi <\theta \le \pi )$

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

$\cos \theta =\frac{\mathbf{A}\mathbf{\cdot }\mathbf{B}}{\Vert \mathbf{A}\Vert \Vert \mathbf{B}\Vert }(-\pi <\theta \le \pi )$

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

$\Vert \mathbf{A}\mathbf{\times }\mathbf{B}{\Vert }^2+(\mathbf{A}\mathbf{\cdot }\mathbf{B}{)}^2=\Vert \mathbf{A}{\Vert }^2\Vert \mathbf{B}{\Vert }^2$

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:

$\cos \alpha =\frac{A_x}{\sqrt{A_x^2+A_y^2+A_z^2}}=\frac{A_x}{\Vert \mathbf{A}\Vert },$

and analogously for angles β, γ. Consequently:

$\mathbf{A}=\Vert \mathbf{A}\Vert (\cos \alpha \hat{\mathbf{i}}+\cos \beta \hat{\mathbf{j}}+\cos \gamma \hat{\mathbf{k}}),$

with $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$ 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:

$\mathrm{\Sigma }=AB\sin \theta ,$

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:

$\mathrm{\Sigma }=\Vert \mathbf{A}\mathbf{\times }\mathbf{B}\Vert =\sqrt{\Vert \mathbf{A}{\Vert }^2\Vert \mathbf{B}{\Vert }^2-(\mathbf{A}\mathbf{\cdot }\mathbf{B}{)}^2}.$

The square of this expression is:

${\mathrm{\Sigma }}^2=(\mathbf{A}\mathbf{\cdot }\mathbf{A})(\mathbf{B}\mathbf{\cdot }\mathbf{B})-(\mathbf{A}\mathbf{\cdot }\mathbf{B})(\mathbf{B}\mathbf{\cdot }\mathbf{A})=\mathrm{\Gamma }(\mathbf{A},\mathbf{B}),$

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

$\mathrm{\Gamma }(\mathbf{A},\mathbf{B})=\left|\begin{array}{cc}\mathbf{A}\mathbf{\cdot }\mathbf{A}& \mathbf{A}\mathbf{\cdot }\mathbf{B}\\ \mathbf{B}\mathbf{\cdot }\mathbf{A}& \mathbf{B}\mathbf{\cdot }\mathbf{B}\end{array}\right|.$

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

$V^2=\mathrm{\Gamma }(\mathbf{A},\mathbf{B},\mathbf{C})=\left|\begin{array}{ccc}\mathbf{A}\mathbf{\cdot }\mathbf{A}& \mathbf{A}\mathbf{\cdot }\mathbf{B}& \mathbf{A}\mathbf{\cdot }\mathbf{C}\\ \mathbf{B}\mathbf{\cdot }\mathbf{A}& \mathbf{B}\mathbf{\cdot }\mathbf{B}& \mathbf{B}\mathbf{\cdot }\mathbf{C}\\ \mathbf{C}\mathbf{\cdot }\mathbf{A}& \mathbf{C}\mathbf{\cdot }\mathbf{B}& \mathbf{C}\mathbf{\cdot }\mathbf{C}\end{array}\right|.$

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.

• $c(\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B}$; distributivity of multiplication by a scalar and addition
• $\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$; commutativity of addition
• $\mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}$; associativity of addition
• $\mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A}$; commutativity of scalar (dot) product
• $\mathbf{A}\times \mathbf{B}=\mathbf{-}\mathbf{B}\times \mathbf{A}$; anticommutativity of vector cross product
• $(\mathbf{A}+\mathbf{B})\cdot \mathbf{C}=\mathbf{A}\cdot \mathbf{C}+\mathbf{B}\cdot \mathbf{C}$; distributivity of addition wrt scalar product
• $(\mathbf{A}+\mathbf{B})\times \mathbf{C}=\mathbf{A}\times \mathbf{C}+\mathbf{B}\times \mathbf{C}$; distributivity of addition wrt vector cross product
• $\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C})=\mathbf{B}\cdot (\mathbf{C}\times \mathbf{A})=\mathbf{C}\cdot (\mathbf{A}\times \mathbf{B})$

$=\left|\begin{array}{ccc}{A}_x& B_x& C_x\\ A_y& B_y& C_y\\ A_z& B_z& C_z\end{array}\right|=[\mathbf{A},\mathbf{B},\mathbf{C}]$ ; scalar triple product

• $\mathbf{A}\mathbf{\times }(\mathbf{B}\times \mathbf{C})=(\mathbf{A}\cdot \mathbf{C})\mathbf{B}-(\mathbf{A}\cdot \mathbf{B})\mathbf{C}$; vector triple product
• $(A\times B)\mathbf{\cdot }(\mathbf{C}\times \mathbf{D})=(\mathbf{A}\cdot \mathbf{C})(\mathbf{B}\cdot \mathbf{D})-(\mathbf{B}\cdot \mathbf{C})(\mathbf{A}\cdot \mathbf{D})$; Binet–Cauchy identity in three dimensions

In particular, when A = C and B = D, the above reduces to:

$\mathbf{(}\mathbf{A}\mathbf{\times }\mathbf{B}\mathbf{)}\mathbf{\cdot }\mathbf{(}\mathbf{A}\mathbf{\times }\mathbf{B}\mathbf{)}=|\mathbf{A}\mathbf{\times }\mathbf{B}{|}^{\mathbf{2}}=\mathbf{(}\mathbf{A}\mathbf{\cdot }\mathbf{A}\mathbf{)}\mathbf{(}\mathbf{B}\mathbf{\cdot }\mathbf{B}\mathbf{)}\mathbf{-}\mathbf{(}\mathbf{A}\mathbf{\cdot }\mathbf{B}{\mathbf{)}}^{\mathbf{2}}$; Lagrange's identity in three dimensions

• $[\mathbf{A},\mathbf{B},\mathbf{C}]\mathbf{D}=(\mathbf{A}\cdot \mathbf{D})(\mathbf{B}\times \mathbf{C})+(\mathbf{B}\cdot \mathbf{D})(\mathbf{C}\times \mathbf{A})+(\mathbf{C}\cdot \mathbf{D})(\mathbf{A}\times \mathbf{B})$
• A vector quadruple product, which is also a vector, can be defined, which satisfies the following identities:

$(\mathbf{A}\times \mathbf{B})\times (\mathbf{C}\times \mathbf{D})=[\mathbf{A},\mathbf{B},\mathbf{D}]\mathbf{C}-[\mathbf{A},\mathbf{B},\mathbf{C}]\mathbf{D}=[\mathbf{A},\mathbf{C},\mathbf{D}]\mathbf{B}-[\mathbf{B},\mathbf{C},\mathbf{D}]\mathbf{A}$

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:

$\mathbf{D}=\frac{\mathbf{D}\mathbf{\cdot }\mathbf{(}\mathbf{B}\mathbf{\times }\mathbf{C}\mathbf{)}}{[\mathbf{A},\mathbf{B},\mathbf{C}]}\mathbf{A}+\frac{\mathbf{D}\mathbf{\cdot }\mathbf{(}\mathbf{C}\mathbf{\times }\mathbf{A}\mathbf{)}}{[\mathbf{A},\mathbf{B},\mathbf{C}]}\mathbf{B}+\frac{\mathbf{D}\mathbf{\cdot }\mathbf{(}\mathbf{A}\mathbf{\times }\mathbf{B}\mathbf{)}}{[\mathbf{A},\mathbf{B},\mathbf{C}]}\mathbf{C}.$

See also

• Vector space
• Geometric algebra