Vector algebra relations

The magnitude of a vector A can be expressed using the dot product:

${\displaystyle \|\mathbf {A} \|^{2}=\mathbf {A\cdot A} }$

In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem:

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

• The Cauchy–Schwarz inequality: ${\displaystyle \mathbf {A\cdot B} \leq \|\mathbf {A} \|\|\mathbf {B} \|}$
• The triangle inequality: ${\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|}$
• The reverse triangle inequality: ${\displaystyle \|\mathbf {A-B} \|\geq {\bigg |}\|\mathbf {A} \|-\|\mathbf {B} \|{\bigg |}}$

The vector product and the scalar product of two vectors define the angle between them, say θ:[1][2]

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

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

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

The Pythagorean trigonometric identity then provides:

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

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:

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

and analogously for angles β, γ. Consequently:

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

with ${\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}}$ unit vectors along the axis directions.

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

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

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:

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

(If A, B are two-dimensional vectors, this is equal to the determinant of the 2 × 2 matrix with rows A, B.) The square of this expression is:[3]

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

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

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

In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A, B, C is given by the Gram determinant of the three vectors:[3]

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

Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product ${\displaystyle \mathrm {det} [\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]=|\mathbf {A} ,\mathbf {B} ,\mathbf {C} |}$ below.

This process can be extended to n-dimensions.

• Commutativity of addition: ${\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }$.
• Commutativity of scalar product: ${\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} }$.
• Anticommutativity of cross product: ${\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A} }$.
• Distributivity of multiplication by a scalar over addition: ${\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} }$.
• Distributivity of scalar product over addition: ${\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} }$.
• Distributivity of vector product over addition: ${\displaystyle (\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} }$.
• Scalar triple product: ${\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )}$${\displaystyle =|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |=\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|}$.
• Vector triple product: ${\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C} }$.
• Jacobi identity: ${\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )=0}$.
• Binet-Cauchy identity: ${\displaystyle \mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)}$.
• Lagrange's identity: ${\displaystyle |\mathbf {A} \times \mathbf {B} |^{2}=(\mathbf {A} \cdot \mathbf {A} )(\mathbf {B} \cdot \mathbf {B} )-(\mathbf {A} \cdot \mathbf {B} )^{2}}$.
• Vector quadruple product:[4][5] ${\displaystyle (\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )\ =\ |\mathbf {A} \,\mathbf {B} \,\mathbf {D} |\,\mathbf {C} \,-\,|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} \ =\ |\mathbf {A} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {B} \,-\,|\mathbf {B} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {A} }$.
• In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as:[6]

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

• Vector space
• Geometric algebra