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Cross Product and Dot Product: Visual explanation
Cross Product and Dot Product: Visual explanation
Published: 2016/02/04
Channel: Physics Videos by Eugene Khutoryansky
The dot product | Magnetic forces, magnetic fields, and Faraday
The dot product | Magnetic forces, magnetic fields, and Faraday's law | Physics | Khan Academy
Published: 2008/08/09
Channel: Khan Academy
Dot products and duality | Essence of linear algebra, chapter 7
Dot products and duality | Essence of linear algebra, chapter 7
Published: 2016/08/24
Channel: 3Blue1Brown
Vectors - The Dot Product
Vectors - The Dot Product
Published: 2008/10/19
Channel: patrickJMT
Calculus 3 Lecture 11.3:  Using the Dot Product
Calculus 3 Lecture 11.3: Using the Dot Product
Published: 2016/01/29
Channel: Professor Leonard
An Introduction to the Dot Product
An Introduction to the Dot Product
Published: 2011/07/10
Channel: patrickJMT
5. Scalar Product or Dot Product (Hindi)
5. Scalar Product or Dot Product (Hindi)
Published: 2016/01/20
Channel: Ignited Minds
Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy
Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy
Published: 2009/10/09
Channel: Khan Academy
Scalar or Dot product :: 1st year Chapter 2 Physics in Urdu
Scalar or Dot product :: 1st year Chapter 2 Physics in Urdu
Published: 2016/04/01
Channel: SWAP Education Portal
Physics - Mechanics: Vectors (12 of 21) Product Of Vectors: Dot Product
Physics - Mechanics: Vectors (12 of 21) Product Of Vectors: Dot Product
Published: 2013/08/07
Channel: Michel van Biezen
Dot Product of Vectors | Video in HINDI | EduPoint
Dot Product of Vectors | Video in HINDI | EduPoint
Published: 2016/08/26
Channel: EduPoint
Using Dot Product to Find the Angle Between Two Vectors
Using Dot Product to Find the Angle Between Two Vectors
Published: 2013/12/11
Channel: Firefly Lectures
Dot Product & Angle Between Vectors
Dot Product & Angle Between Vectors
Published: 2012/02/08
Channel: ProfRobBob
Vectors : Scalar or Dot Product : ExamSolutions
Vectors : Scalar or Dot Product : ExamSolutions
Published: 2011/08/13
Channel: ExamSolutions
Dot Product
Dot Product
Published: 2009/09/20
Channel: lasseviren1
Dot Product : Find Angle Between Two Vectors , Another Example
Dot Product : Find Angle Between Two Vectors , Another Example
Published: 2012/09/27
Channel: patrickJMT
Meaning of dot product
Meaning of dot product
Published: 2013/09/21
Channel: Shridhar Jagtap
Calculus III: The Dot Product (Level 11 of 12) | Work, Examples VIII
Calculus III: The Dot Product (Level 11 of 12) | Work, Examples VIII
Published: 2016/05/15
Channel: Math Fortress
Physics Part I  Chapter 2 Scalar or Dot Product
Physics Part I Chapter 2 Scalar or Dot Product
Published: 2016/10/12
Channel: PGC Lectures
Linear Algebra 20g: The Dot Product - One of the Most Brilliant Ideas in All of Linear Algebra
Linear Algebra 20g: The Dot Product - One of the Most Brilliant Ideas in All of Linear Algebra
Published: 2015/02/18
Channel: MathTheBeautiful
Class 12 Maths : Vectors - Dot product ( Scalar product ) Part 1
Class 12 Maths : Vectors - Dot product ( Scalar product ) Part 1
Published: 2014/12/11
Channel: studyezee
Dot product (Scalar Product ) of two vectors| CBSE 12 Maths NCERT 10.3 intro
Dot product (Scalar Product ) of two vectors| CBSE 12 Maths NCERT 10.3 intro
Published: 2017/01/09
Channel: cbseclass videos
Dot vs. cross product | Physics | Khan Academy
Dot vs. cross product | Physics | Khan Academy
Published: 2008/08/09
Channel: Khan Academy
Dot Product Intuition | BetterExplained
Dot Product Intuition | BetterExplained
Published: 2017/08/01
Channel: Better Explained
Dot Product in vectors -JEE|Medical|CBSE|Hindi
Dot Product in vectors -JEE|Medical|CBSE|Hindi
Published: 2016/12/16
Channel: Any Time Padhai Academy
6.5: Vectors: The Dot Product - The Nature of Code
6.5: Vectors: The Dot Product - The Nature of Code
Published: 2015/08/08
Channel: The Coding Train
Dot Product of Two Vectors Explained, Parallel, Perpendicular, Neither, Physics & Precalculus
Dot Product of Two Vectors Explained, Parallel, Perpendicular, Neither, Physics & Precalculus
Published: 2017/04/27
Channel: The Organic Chemistry Tutor
Scalar or Dot Product
Scalar or Dot Product
Published: 2017/07/15
Channel: Punjab Group Of Colleges
Scalar Product (Dot Product) Part 1 - IIT JEE Main and Advanced Maths Video Lecture
Scalar Product (Dot Product) Part 1 - IIT JEE Main and Advanced Maths Video Lecture
Published: 2014/06/28
Channel: Rao IIT Academy
Dot and Cross Products
Dot and Cross Products
Published: 2012/08/31
Channel: jg394
Inner Product is Dot Product, Turned on Its Head
Inner Product is Dot Product, Turned on Its Head
Published: 2017/01/25
Channel: MathTheBeautiful
Dot product and angle between two vectors proof
Dot product and angle between two vectors proof
Published: 2016/10/11
Channel: Matthew James
Calculus III: The Dot Product (Level 1 of 12) | Geometric Definition
Calculus III: The Dot Product (Level 1 of 12) | Geometric Definition
Published: 2016/02/17
Channel: Math Fortress
Scalar Product of Two Vectors - Class 12
Scalar Product of Two Vectors - Class 12
Published: 2015/05/27
Channel: Uniclass Content
Intro to the Vector Dot Product: How to code a Field of View in Godot
Intro to the Vector Dot Product: How to code a Field of View in Godot
Published: 2017/06/10
Channel: GDquest
Calculating dot and cross products with unit vector notation | Physics | Khan Academy
Calculating dot and cross products with unit vector notation | Physics | Khan Academy
Published: 2008/08/09
Channel: Khan Academy
[2015] Statics 07: Dot Product of Cartesian Vectors[with closed caption]
[2015] Statics 07: Dot Product of Cartesian Vectors[with closed caption]
Published: 2015/01/20
Channel: Yiheng Wang
The Scalar Product or Dot Product for Physics
The Scalar Product or Dot Product for Physics
Published: 2009/09/20
Channel: lasseviren1
Applications of the Dot Product and Cross Product
Applications of the Dot Product and Cross Product
Published: 2017/04/08
Channel: AlRichards314
Index/Tensor Notation: The scalar or dot product - Lesson 2
Index/Tensor Notation: The scalar or dot product - Lesson 2
Published: 2016/10/24
Channel: WeSolveThem
7.2 Dot (Scalar) Product
7.2 Dot (Scalar) Product
Published: 2013/03/28
Channel: TheDovoin
Proving vector dot product properties | Vectors and spaces | Linear Algebra | Khan Academy
Proving vector dot product properties | Vectors and spaces | Linear Algebra | Khan Academy
Published: 2009/10/10
Channel: Khan Academy
Dot Product and the Law of Cosines
Dot Product and the Law of Cosines
Published: 2012/09/18
Channel: Jason Rose
Proving Dot Product Formula
Proving Dot Product Formula
Published: 2014/02/23
Channel: Kibblesnbits
Applications of the Dot Product.avi
Applications of the Dot Product.avi
Published: 2011/01/19
Channel: AlRichards314
The Triple Scalar Product
The Triple Scalar Product
Published: 2010/12/28
Channel: Mathispower4u
Inner Product and Orthogonal Functions , Quick Example
Inner Product and Orthogonal Functions , Quick Example
Published: 2012/10/29
Channel: patrickJMT
Inner Product & Inner Product Space - Introduction
Inner Product & Inner Product Space - Introduction
Published: 2013/09/04
Channel: ASTROTZUR
Matlab   Sect 25   Calculating the Vector Dot Product and Cross Product
Matlab Sect 25 Calculating the Vector Dot Product and Cross Product
Published: 2012/08/29
Channel: CodeCodeable
Linear Algebra 20j: The Dot Product, Matrix Multiplication, and the Magic of Orthogonal Matrices
Linear Algebra 20j: The Dot Product, Matrix Multiplication, and the Magic of Orthogonal Matrices
Published: 2015/02/18
Channel: MathTheBeautiful
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product space.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths).

The name "dot product" is derived from the centered dot· " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, which is the case for the vector product in three-dimensional space.

Definition[edit]

The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.

In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space Rn. In such a presentation, the notions of length and angles are not primitive.[clarification needed] They are defined by means of the dot product: the length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle of two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.[citation needed]

Algebraic definition[edit]

The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:[1]

where Σ denotes summation notation and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors [1, 3, −5] and [4, −2, −1] is:

The dot product can also be written as:

.

Here, means the transpose of .

Using the above example, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get the result (1 × 1 matrix is obtained by matrix multiplication, which is a scalar):

.

Geometric definition[edit]

In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction that the arrow points. The magnitude of a vector a is denoted by . The dot product of two Euclidean vectors a and b is defined by[2][3]

where θ is the angle between a and b.

In particular, if a and b are orthogonal, then the angle between them is 90° and

At the other extreme, if they are codirectional, then the angle between them is 0° and

This implies that the dot product of a vector a with itself is

which gives

the formula for the Euclidean length of the vector.

Scalar projection and first properties[edit]

Scalar projection

The scalar projection (or scalar component) of a Euclidean vector a in the direction of a Euclidean vector b is given by

where θ is the angle between a and b.

In terms of the geometric definition of the dot product, this can be rewritten

where is the unit vector in the direction of b.

Distributive law for the dot product

The dot product is thus characterized geometrically by[4]

The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar α,

It also satisfies a distributive law, meaning that

These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that is never negative and is zero if and only if

Equivalence of the definitions[edit]

If e1, ..., en are the standard basis vectors in Rn, then we may write

The vectors ei are an orthonormal basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length

and since they form right angles with each other, if ij,

Thus in general we can say that:

Where δ ij is the Kronecker delta.

Also, by the geometric definition, for any vector ei and a vector a, we note

where ai is the component of vector a in the direction of ei.

Now applying the distributivity of the geometric version of the dot product gives

which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.

Properties[edit]

The dot product fulfills the following properties if a, b, and c are real vectors and r is a scalar.[1][2]

  1. Commutative:
    which follows from the definition (θ is the angle between a and b):
  2. Distributive over vector addition:
  3. Bilinear:
  4. Scalar multiplication:
  5. Not associative because the dot product between a scalar (a ⋅ b) and a vector (c) is not defined, which means that the expressions involved in the associative property, (a ⋅ b) ⋅ c or a ⋅ (b ⋅ c) are both ill-defined.[5] Note however that the previously mentioned scalar multiplication property is sometimes called the "associative law for scalar and dot product"[6] or one can say that "the dot product is associative with respect to scalar multiplication" because c (ab) = (c a) ⋅ b = a ⋅ (c b).[7]
  6. Orthogonal:
    Two non-zero vectors a and b are orthogonal if and only if ab = 0.
  7. No cancellation:
    Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law:
    If ab = ac and a0, then we can write: a ⋅ (bc) = 0 by the distributive law; the result above says this just means that a is perpendicular to (bc), which still allows (bc) ≠ 0, and therefore bc.
  8. Product Rule: If a and b are functions, then the derivative (denoted by a prime ′) of ab is a′ ⋅ b + ab.

Application to the law of cosines[edit]

Triangle with vector edges a and b, separated by angle θ.

Given two vectors a and b separated by angle θ (see image right), they form a triangle with a third side c = ab. The dot product of this with itself is:

which is the law of cosines.

Triple product expansion[edit]

This is an identity (also known as Lagrange's formula) involving the dot- and cross-products. It is written as:[1][2]

which may be remembered as "BAC minus CAB", keeping in mind which vectors are dotted together. This formula finds application in simplifying vector calculations in physics.

Physics[edit]

In physics,vector magnitude is a scalar in the physical sense, i.e. a physical quantity independent of the coordinate system, expressed as the product of a numerical value and a physical unit, not just a number. The dot product is also a scalar in this sense, given by the formula, independent of the coordinate system. Examples include:[8][9]

Generalizations[edit]

Complex vectors[edit]

For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance the dot product of a vector with itself would be an arbitrary complex number, and could be zero without the vector being the zero vector (such vectors are called isotropic); this in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the scalar product, through the alternative definition[1]

where bi is the complex conjugate of bi. Then the scalar product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However this scalar product is thus sesquilinear rather than bilinear: it is conjugate linear and not linear in b, and the scalar product is not symmetric, since

The angle between two complex vectors is then given by

This type of scalar product is nevertheless useful, and leads to the notions of Hermitian form and of general inner product spaces.

Inner product[edit]

The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers or the field of complex numbers . It is usually denoted using angular brackets by .

The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.

Functions[edit]

The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length-n vector u is, then, a function with domain {k ∈ ℕ ∣ 1 ≤ kn}, and ui is a notation for the image of i by the function/vector u.

This notion can be generalized to continuous functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some interval axb (also denoted [a, b]):[1]

Generalized further to complex functions ψ(x) and χ(x), by analogy with the complex inner product above, gives[1]

Weight function[edit]

Inner products can have a weight function, i.e. a function which weights each term of the inner product with a value.

Dyadics and matrices[edit]

Matrices have the Frobenius inner product, which is analogous to the vector inner product. It is defined as the sum of the products of the corresponding components of two matrices A and B having the same size:

(For real matrices)

Dyadics have a dot product and "double" dot product defined on them, see Dyadics (Product of dyadic and dyadic) for their definitions.

Tensors[edit]

The inner product between a tensor of order n and a tensor of order m is a tensor of order n + m − 2, see tensor contraction for details.

Computation[edit]

Algorithms[edit]

The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. To avoid this, approaches such as the Kahan summation algorithm are used.

Libraries[edit]

A dot product function is included in BLAS level 1.

See also[edit]

Notes[edit]

  1. ^ The term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.[citation needed]

References[edit]

  1. ^ a b c d e f S. Lipschutz; M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1. 
  2. ^ a b c M.R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 978-0-07-161545-7. 
  3. ^ A I Borisenko; I E Taparov (1968). Vector and tensor analysis with applications. Translated by Richard Silverman. Dover. p. 14. 
  4. ^ Arfken, G. B.; Weber, H. J. (2000). Mathematical Methods for Physicists (5th ed.). Boston, MA: Academic Press. pp. 14–15. ISBN 978-0-12-059825-0. .
  5. ^ Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DotProduct.html
  6. ^ T. Banchoff; J. Wermer (1983). Linear Algebra Through Geometry. Springer Science & Business Media. p. 12. ISBN 978-1-4684-0161-5. 
  7. ^ A. Bedford; Wallace L. Fowler (2008). Engineering Mechanics: Statics (5th ed.). Prentice Hall. p. 60. ISBN 978-0-13-612915-8. 
  8. ^ K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering (3rd ed.). Cambridge University Press. ISBN 978-0-521-86153-3. 
  9. ^ M. Mansfield; C. O’Sullivan (2011). Understanding Physics (4th ed.). John Wiley & Sons. ISBN 978-0-47-0746370. 

External links[edit]

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