Dot product

The dot product ( inner product also rarely dot product ) is a mathematical operation, the two vectors is a number ( scalar) assigns. It is an object of the analytic geometry and linear algebra. Historically, it was first introduced in the Euclidean space. Geometrically, we can calculate the dot product of two vectors and in three-dimensional space of intuition, according to the formula

This call and the respective lengths (amounts ) of the vectors. With the cosine of the angle enclosed by the two vectors is referred to.

Applies in a Cartesian coordinate system

Knowing the Cartesian coordinates of the vectors, so you can calculate with this formula, the scalar product and the above formula then the angle between the two vectors.

In linear algebra, this concept is generalized. An inner product is here a function which maps two elements of a real or complex vector space a scalar. In general no scalar product is defined in a vector space from the outset. A space with a scalar product is called the inner product space or pre-Hilbert space. These vector spaces generalize Euclidean space and thus enable the application of geometric methods to abstract structures.

• 2.1 Definition ( axiomatic )
• 2.2 Examples 2.2.1 standard scalar product in Rn and Cn
• 2.2.2 General scalar products in Rn and Cn
• 2.2.3 L2 - scalar product for functions
• 2.2.4 Frobenius inner product of matrices

In Euclidean space

Geometric Definition and notation

Vectors in three-dimensional Euclidean space or in the two-dimensional Euclidean plane can be represented as arrows. Here, arrows that are parallel, of equal length and have the same orientation, the same vector Represents the scalar product of two vectors and is a scalar, that is a real number. Geometrically, it can be defined as follows:

, Respectively, and the lengths of the vectors and and the designated angle, and enclosed by, then

As with the normal multiplication, but less than there, the multiplication sign is sometimes omitted when it is clear what is meant:

Instead you write in this case, sometimes

Other common notations and

Illustration

To illustrate, the definition, we consider the orthogonal projection of the vector on the direction and is given by

It is then and for the scalar product of and is:

This relationship is also sometimes used for the definition of the scalar product.

Examples

In all three examples and applies. The scalar result using the special cosines, or:

And at a 60 ° angle

And orthogonal

In Cartesian coordinates,

If one introduces into the Euclidean plane or in Euclidean space Cartesian coordinates, so each vector has a coordinate representation as a 2 - or 3- tuple, which is usually written as a column.

In the Euclidean plane is then obtained for the scalar product of the vectors

The representation

For the canonical unit vectors and your motto is

It follows ( in anticipation of explained below properties of the scalar product ):

In three-dimensional Euclidean space is obtained according to the vectors

The representation

For example, the dot product of two vectors is calculated

As follows:

Properties

From the geometric definition gives rise directly to:

• Are parallel and have the same orientation ( ), then
• In particular, the scalar product of a vector with itself gives the square of its length:
• Are parallel and oppositely oriented ( ), then
• Are and orthogonal ( ), then
• Is an acute angle, the following applies
• Is an obtuse angle, the following applies

As a function which assigns to each ordered pair of vectors, the real number, the scalar product has the following features that you would expect from a multiplication:

The properties of 2 and 3 we group together to: The dot product is bilinear.

The term " mixed associative law " for the second characteristic shows that this is a scalar, and two vectors are linked so that the staples can be exchanged as in the associative law. Since the scalar product is no inner join, a scalar product of three vectors is not defined, so the question of a real associativity does not arise. In the expression, only the first multiplier is a scalar product of two vectors, the second is the product of a scalar to a vector (S multiplication ). The term represents a vector, a multiple of the vector In contrast, the expression is a multiple of, as in general so true

Neither the geometrical definition still defined in the Cartesian coordinate is arbitrary. Both follow from the geometrically motivated requirement that the scalar product of a vector with itself is the square of its length, and the algebraically motivated requirement that the inner product satisfies the above properties 1-3.

Amount of vectors and included angle

By means of the scalar product, it is possible to calculate from the coordinates representation, the length ( the amount ) of a vector:

For vectors of the two-dimensional space

One recognizes here the Pythagorean Theorem again. In three-dimensional space is considered in accordance with

By combining the geometric definition of the coordinate representation, can be calculated from the coordinates of two vectors the angle enclosed by them. from

Follows

The lengths of the two vectors

Be so

The cosine of the angle enclosed by the two vectors is calculated as

Is therefore

Orthogonality and orthogonal projection

Two vectors and are orthogonal if and only if their dot product is zero, ie

The orthogonal projection of the given by the vector direction is the vector with

So

The projection of the vector, the end point of the nadir point to the end point of which is given by line through the origin. The vector is perpendicular to

Is a unit vector (ie is ), then the formula simplifies to

Respect to the cross product

Another way in which two vectors and multiplication to combine with each other in three-dimensional space, the exterior product or cross product is In contrast to the scalar product of the result of the cross product is not a scalar, but again a vector. This vector is perpendicular to that of the two factors, and the clamped level, and its length corresponds to the area of ​​the parallelogram, which is spanned by these.

For the connection of cross and dot product of the following calculation rules apply:

The combination of cross product and dot product of the first two rules are also called scalar triple product; it makes the volume of the oriented plane defined by the three vectors parallelepiped.

Applications

In geometry

The scalar product allows complex sentences in which angles of the speech, is easy to prove.

Claim: ( cosine )

Proof: With the help of the indicated vectors follows (. , The direction of is irrelevant ) squaring yields

And thus

In Physics

In physics are many variables, such as, for example, the job defined by scalar:

With the vectorial quantities force and displacement. This is the angle between the direction of force and the direction of the path. With the component of the force in the direction of the path is denoted by the component of the path in the direction of the force.

Example: A car of weight transported over an inclined plane from to. The lifting operation is calculated as

In general vector spaces

Take the above properties as an opportunity to generalize the notion of the scalar product on arbitrary real and complex vector spaces. A scalar product is then a function that assigns a body member ( scalar) two vectors and satisfies the abovementioned properties. In the complex case is modified so to have the symmetry and bilinearity to save the Positivdefinitheit (which is never fulfilled for complex symmetric bilinear forms ).

In the general theory of the variables of vectors, that is, elements of an arbitrary vector space, generally not indicated by arrows. The scalar product is not usually designated by a Malpunkt, but by a pair of angle brackets. So for the scalar product of vectors and to write. Other common notations are (especially in quantum mechanics in the form of the Bra- Ket notation ), and.

Definition ( axioms )

A scalar or inner product on a real vector space is a positive definite symmetric bilinear form that is for and the following conditions apply:

A scalar or inner product on a complex vector space is a positive definite hermitian sesquilinear that is for and the following conditions apply:

• ( semilinear in the first argument )
• (linear in the second argument )

A real or complex vector space, in which a scalar product is defined is called scalar product or pre-Hilbert space. A finite dimensional real vector space with scalar product is also called Euclidean vector space, in the complex case one speaks of a unitary vector space. Accordingly, the scalar product is sometimes referred to in the Euclidean vector space as a Euclidean scalar product, in a unitary vector space as a unitary scalar product. The term " Euclidean scalar product " but is also used specifically for the above described geometric scalar product or the standard scalar described below in.

• Often, each and every symmetric bilinear Hermitian sesquilinear form is called a scalar product; with this use of language to describe the above definitions, positive definite scalar products.
• The two specified axiom systems are not minimal. In the real case due to the symmetry follows the linearity in the first argument from the linearity in the second argument (and vice versa). Similarly, follows in the complex case due to the hermeticity the semi -linearity in the first argument from the linearity in the second argument (and vice versa).
• In the complex case, the scalar product is often an alternative, namely to be linear in the first and semilinear defined in the second argument. This version tends to occur in mathematics and in particular in the analysis, while in physics mainly the above version is used (see Bra and ket vectors ). The difference between the two versions is in the effect of the scalar multiplication with regard to the homogeneity. After the alternate version is valid for and and. The additivity is understood the same in both versions. Similarly, the standards obtained from the scalar product in both versions are identical.
• A pre-Hilbert space, which is complete with respect to the norm induced by the inner product is referred to as the Hilbert space.

Examples

Standard scalar product in Rn and Cn

Starting from the presentation of the Euclidean scalar product is defined in Cartesian coordinates in the linear algebra in the standard scalar -dimensional coordinate space for by

The above treated " geometric " scalar product in Euclidean space corresponds to the special case of the case of the - dimensional complex vector space we define the standard scalar for by

Where the overline denotes the complex conjugation. In mathematics, the alternative version is often common, wherein the second argument is conjugated instead of the first.

The standard scalar product in and can be written as a matrix product by the vector as a matrix ( column vector ) interpreted applies in the real case

Where the row vector that emerges from the column vector by transposing. In the complex case (for the semilinear left, right linear case )

Where the Hermitian adjoint to row vector is.

General scalar products in Rn and Cn

More generally defined in the real case every symmetric and positive definite matrix over

A scalar product; just as in the complex case is over, for each positive definite Hermitian matrix

A scalar product defined. Here denote the angle brackets on the right side of the standard scalar product, the angle brackets with the index A on the left side of the image defined by the dot product matrix.

Each inner product on or can be represented in this way by a positive definite symmetric matrix (or positive definite Hermitian matrix).

L2 - scalar product for functions

In the infinite- dimensional vector space of continuous real-valued functions on the interval, the dot product is by

Defined for all.

For generalizations of this example, see pre-Hilbert space and Hilbert space.

Frobenius inner product of matrices

On the die space of real matrices is for by

A scalar product defined. Accordingly, on the space of complex matrices for by

A scalar product defined. This scalar product is called the Frobenius inner product and the corresponding norm is Frobenius norm.

Standard, angles and orthogonality

The length of a vector in Euclidean space corresponds in general Skalarprodukträumen the scalar product induced by the norm. We define this standard, by transferring the formula for the length of the Euclidean space, as the root of the scalar product of the vector with itself:

This is possible because, due to the positive definiteness is not negative. The required than standard axiom triangle inequality follows here from the Cauchy- Schwarz inequality

If so, this inequality can be

Be reshaped. Therefore, in general, can also be real vector spaces by means of

Define the angle between two vectors. The so- defined angle is between 0 ° and 180 °, ie between 0 and For angles between complex vectors, there are a number of different definitions.

In the general case known as vectors, the inner product is zero, orthogonal:

Matrix representation

Is one - dimensional vector space, and a base of each of may dot on by a ( ) matrix the Gram matrix of the dot product will be described. Your entries are the scalar products of the basis vectors:

The scalar product can then be represented using the base: Do the vectors with respect to the base, the representation

Shall apply in the real case

If we denote by the coordinate vectors

So it shall

Wherein the matrix product of a matrix provides, therefore, a real number. With the line vector is referred to, which is produced by transposition of the column vector.

In the complex case, according to

Where the overline denotes complex conjugation and adjoint to row vector is.

Is an orthonormal basis, that is, is valid for all and for all as is the identity matrix, and it is

In the real case or

In the complex case. With respect to an orthonormal basis corresponds to the dot product of and that is the standard scalar product of vectors and coordinate or