Vector algebra relations

The relations below apply to vectors in a three-dimensional Euclidean space.[1] 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).

See also: Vector calculus identities

Magnitudes

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

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

The magnitude also can be expressed using the dot product:

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

Inequalities

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

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

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

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

Here the notation A × B denotes the vector cross product of vectors A and B. 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.

Areas and volumes

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.

Addition and multiplication of vectors

Some of the following algebraic relations refer to the dot product and the cross product of vectors.[1]

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

• ${\displaystyle |\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} \ =\ \left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)\,+\,\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)\,+\,\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)}$
• ${\displaystyle \!(\mathbf {A} \times \mathbf {B} )\times (\mathbf {C} \times \mathbf {D} )=\left(\mathbf {A} \cdot (\mathbf {B} \times \mathbf {D} )\right)\mathbf {C} -\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D} }$ (vector quadruple product)[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 a basis {A,B,C} as:[6]

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

See also

• Vector space
• Geometric algebra

References

1. See, for example, Lyle Frederick Albright (2008). "§2.5.1 Vector algebra". Albright's chemical engineering handbook. CRC Press. p. 68. ISBN 978-0-8247-5362-7.
2. Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.
3. Richard Courant, Fritz John (2000). "Areas of parallelograms and volumes of parallelepipeds in higher dimensions". Introduction to calculus and analysis, Volume II (Reprint of original 1974 Interscience ed.). Springer. pp. 190–195. ISBN 3-540-66569-2.
4. Vidwan Singh Soni (2009). "§1.10.2 Vector quadruple product". Mechanics and relativity. PHI Learning Pvt. Ltd. pp. 11–12. ISBN 978-81-203-3713-8.
5. This formula is applied to spherical trigonometry by Edwin Bidwell Wilson, Josiah Willard Gibbs (1901). "§42 in Direct and skew products of vectors". Vector analysis: a text-book for the use of students of mathematics. Scribner. pp. 77ff.
6. Joseph George Coffin (1911). Vector analysis: an introduction to vector-methods and their various applications to physics and mathematics (2nd ed.). Wiley. p. 56.