Vector algebra relations

See also: Vector calculus identities

The relations below apply to vectors in a three-dimensional Euclidean space. Some, but not all of them, extend to vectors of higher dimensions. In particular, the cross product of vectors is defined only in three dimensions (but see Seven-dimensional cross product).

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:

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

(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 $\mathrm {det} [\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]=|\mathbf {A} ,\mathbf {B} ,\mathbf {C} |$ below.

This process can be extended to n-dimensions.

Addition and multiplication of vectors

Some of the following algebraic relations refer to the dot product and the cross product of vectors.

• $\mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A}$; commutativity of addition
• $\mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A}$; commutativity of scalar product
• $\mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A}$; anticommutativity of vector product
• $c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B}$; distributivity of multiplication by a scalar over addition
• $\left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C}$; distributivity of scalar product over addition
• $(\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C}$; distributivity of vector product over addition
• $\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} |=\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|$ (scalar triple product)
• $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C}$ (vector triple product)
• $(\mathbf {A} \times \mathbf {B} )\times \mathbf {C} =(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {B} \cdot \mathbf {C} )\mathbf {A}$ (vector triple product)
• $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \times \mathbf {B} )\times \mathbf {C} +\mathbf {B} \times (\mathbf {A} \times \mathbf {C} )$ (Jacobi identity)
• $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )=0$ (Jacobi identity)
• $\!(\mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} ))\,\mathbf {D} =(\mathbf {A} \cdot \mathbf {D} )\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)$
• $\mathbf {\left(A\times B\right)\cdot } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)$; Binet–Cauchy identity in three dimensions
• $|\mathbf {A} \times \mathbf {B} |^{2}=(\mathbf {A} \cdot \mathbf {A} )(\mathbf {B} \cdot \mathbf {B} )-(\mathbf {A} \cdot \mathbf {B} )^{2}$; Lagrange's identity in three dimensions

• $|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} \ =\ \left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)\,+\,\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)\,+\,\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)$
• $\!(\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=\left(\mathbf {A} \cdot (\mathbf {B} \times \mathbf {D} )\right)\mathbf {C} -\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D}$ (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 a basis {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