Vector algebra relations

See also: Vector calculus identities

The following are important identities in vector algebra. Identities that involve the magnitude of a vector $\|\mathbf {A} \|$, or the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension. Identities that use the cross product (vector product) A×B are defined only in three dimensions. (There is a seven-dimensional cross product, but the identities do not hold in seven dimensions.)

Magnitudes

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

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

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

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

Inequalities

• The Cauchy–Schwarz inequality: $\mathbf {A} \cdot \mathbf {B} \leq \left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|$
• The triangle inequality: $\|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|$
• The reverse triangle inequality: $\|\mathbf {A-B} \|\geq {\Bigl |}\|\mathbf {A} \|-\|\mathbf {B} \|{\Bigr |}$

Angles

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

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

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

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

The Pythagorean trigonometric identity then provides:

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

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:

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

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:

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

(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:

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

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 parallelepiped spanned by the three vectors A, B, C is given by the Gram determinant of the three vectors:

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

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

This process can be extended to n-dimensions.

Addition and multiplication of vectors

• Commutativity of addition: $\mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A}$.
• Commutativity of scalar product: $\mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A}$.
• Anticommutativity of cross product: $\mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A}$.
• Distributivity of multiplication by a scalar over addition: $c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B}$.
• Distributivity of scalar product over addition: $\left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C}$.
• Distributivity of vector product over addition: $(\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C}$.
• Scalar triple 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} )$$=|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |={\begin{vmatrix}A_{x}&B_{x}&C_{x}\\A_{y}&B_{y}&C_{y}\\A_{z}&B_{z}&C_{z}\end{vmatrix}}$.
• Vector triple product: $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C}$.
• Jacobi identity: $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )=\mathbf {0}$.
• Binet-Cauchy identity: $\mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)$.
• Lagrange's identity: $|\mathbf {A} \times \mathbf {B} |^{2}=(\mathbf {A} \cdot \mathbf {A} )(\mathbf {B} \cdot \mathbf {B} )-(\mathbf {A} \cdot \mathbf {B} )^{2}$.
• Vector quadruple product: $(\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )\ =\ |\mathbf {A} \,\mathbf {B} \,\mathbf {D} |\,\mathbf {C} \,-\,|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} \ =\ |\mathbf {A} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {B} \,-\,|\mathbf {B} \,\mathbf {C} \,\mathbf {D} |\,\mathbf {A}$.
• In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as:$$\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} .$$

See also

• Vector space
• Geometric algebra