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:

$\|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\$

The magnitude also can be expressed using the dot product:

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

Inequalities

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

$\|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|$; the triangle inequality in three dimensions

$\|\mathbf {A-B} \|\geq \|\mathbf {A} \|-\|\mathbf {B} \|$; 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:

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

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

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

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

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

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:

$\cos \alpha ={\frac {A_{x}}{\sqrt {A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}}={\frac {A_{x}}{\|\mathbf {A} \|}}\ ,$

and analogously for angles β, γ. Consequently:

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

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:

$\Sigma =AB\ \sin \theta \ ,$

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:

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

The square of this expression is:

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

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

$\Gamma (\mathbf {A} ,\ \mathbf {B} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} \end{vmatrix}}\ .$

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:

$V^{2}=\Gamma (\mathbf {A} ,\ \mathbf {B} ,\ \mathbf {C} )={\begin{vmatrix}\mathbf {A\cdot A} &\mathbf {A\cdot B} &\mathbf {A\cdot C} \\\mathbf {B\cdot A} &\mathbf {B\cdot B} &\mathbf {B\cdot C} \\\mathbf {C\cdot A} &\mathbf {C\cdot B} &\mathbf {C\cdot C} \end{vmatrix}}\ .$

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
• $\mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A}$; anticommutativity of vector cross product
• $\left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C}$; distributivity of addition wrt scalar product
• $\left(\mathbf {A} +\mathbf {B} \right)\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C}$; distributivity of addition wrt vector cross product
• $\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)=\mathbf {B} \cdot \left(\mathbf {C} \times \mathbf {A} \right)=\mathbf {C} \cdot \left(\mathbf {A} \times \mathbf {B} \right)$

$=\left|{\begin{array}{ccc}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{array}}\right|=[\mathbf {A,\ B,\ C} ]$ ; scalar triple product

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

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

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

• $[\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)$
• 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:

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

See also

• Vector space
• Geometric algebra