The following are important identities in
vector algebra. Identities that involve the magnitude of a vector
, or the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
(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.
Magnitudes
The magnitude of a vector A can be expressed using the dot product:
:
In three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, the magnitude of a vector is determined from its three components using
Pythagoras' theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
:
:
Inequalities
*The
Cauchy–Schwarz inequality:
*The
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
:
*The
reverse triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
:
Angles
The vector product and the scalar product of two vectors define the angle between them, say ''θ'':
[
]
:
To satisfy the right-hand rule, for positive ''θ'', vector B is counter-clockwise from A, and for negative ''θ'' it is clockwise.
:
The Pythagorean trigonometric identity then provides:
:
If a vector A = (''Ax, Ay, Az'') makes angles ''α'', ''β'', ''γ'' with an orthogonal set of ''x-'', ''y-'' and ''z-''axes, then:
:
and analogously for angles β, γ. Consequently:
:
with 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:
:
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:
:
(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:
:
where Γ(A, B) is the Gram determinant of A and B defined by:
:
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:[
:
Since A, B, C are three-dimensional vectors, this is equal to the square of the ]scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
below.
This process can be extended to ''n''-dimensions.
Addition and multiplication of vectors
* Commutativity of addition: .
* Commutativity of scalar product: .
* Anticommutativity
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
of cross product: .
* Distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
of multiplication by a scalar over addition: .
* Distributivity of scalar product over addition: .
* Distributivity of vector product over addition: .
* Scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
:
* Vector triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
: .
* Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
:
* Binet-Cauchy identity:
* Lagrange's identity: .
* Vector quadruple product:[This formula is applied to spherical trigonometry by ]
* A consequence of the previous equation:
*In 3 dimensions, a vector D can be expressed in terms of basis vectors
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
as:
See also
*Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
* Geometric algebra
Notes
References
{{Reflist
Mathematical identities
Mathematics-related lists