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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's law | Physics | Khan Academy

Published: 2008/08/09

Channel: Khan Academy

Dot products and duality | Essence of linear algebra, chapter 7

Published: 2016/08/24

Channel: 3Blue1Brown

Vectors - The Dot Product

Published: 2008/10/19

Channel: patrickJMT

Calculus 3 Lecture 11.3: Using the Dot Product

Published: 2016/01/29

Channel: Professor Leonard

An Introduction to the Dot Product

Published: 2011/07/10

Channel: patrickJMT

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

Published: 2009/10/09

Channel: Khan Academy

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

Published: 2013/08/07

Channel: Michel van Biezen

Dot Product of Vectors | Video in HINDI | EduPoint

Published: 2016/08/26

Channel: EduPoint

Using Dot Product to Find the Angle Between Two Vectors

Published: 2013/12/11

Channel: Firefly Lectures

Dot Product & Angle Between Vectors

Published: 2012/02/08

Channel: ProfRobBob

Vectors : Scalar or Dot Product : ExamSolutions

Published: 2011/08/13

Channel: ExamSolutions

Dot Product

Published: 2009/09/20

Channel: lasseviren1

Dot Product : Find Angle Between Two Vectors , Another Example

Published: 2012/09/27

Channel: patrickJMT

Meaning of dot product

Published: 2013/09/21

Channel: Shridhar Jagtap

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

Published: 2016/10/12

Channel: PGC Lectures

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

Published: 2014/12/11

Channel: studyezee

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

Published: 2008/08/09

Channel: Khan Academy

Dot Product Intuition | BetterExplained

Published: 2017/08/01

Channel: Better Explained

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

Published: 2015/08/08

Channel: The Coding Train

Dot Product of Two Vectors Explained, Parallel, Perpendicular, Neither, Physics & Precalculus

Published: 2017/04/27

Channel: The Organic Chemistry Tutor

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

Published: 2014/06/28

Channel: Rao IIT Academy

Dot and Cross Products

Published: 2012/08/31

Channel: jg394

Inner Product is Dot Product, Turned on Its Head

Published: 2017/01/25

Channel: MathTheBeautiful

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

Published: 2016/02/17

Channel: Math Fortress

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

Published: 2017/06/10

Channel: GDquest

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]

Published: 2015/01/20

Channel: Yiheng Wang

The Scalar Product or Dot Product for Physics

Published: 2009/09/20

Channel: lasseviren1

Applications of the Dot Product and Cross Product

Published: 2017/04/08

Channel: AlRichards314

Index/Tensor Notation: The scalar or dot product - Lesson 2

Published: 2016/10/24

Channel: WeSolveThem

7.2 Dot (Scalar) Product

Published: 2013/03/28

Channel: TheDovoin

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

Published: 2012/09/18

Channel: Jason Rose

Proving Dot Product Formula

Published: 2014/02/23

Channel: Kibblesnbits

Applications of the Dot Product.avi

Published: 2011/01/19

Channel: AlRichards314

The Triple Scalar Product

Published: 2010/12/28

Channel: Mathispower4u

Inner Product and Orthogonal Functions , Quick Example

Published: 2012/10/29

Channel: patrickJMT

Inner Product & Inner Product Space - Introduction

Published: 2013/09/04

Channel: ASTROTZUR

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

Published: 2015/02/18

Channel: MathTheBeautiful

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.

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 **R**^{n}. 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]}

The dot product of two vectors **a** = [*a*_{1}, *a*_{2}, ..., *a*_{n}] and **b** = [*b*_{1}, *b*_{2}, ..., *b*_{n}] 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):

- .

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.

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**.

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

If **e**_{1}, ..., **e**_{n} are the standard basis vectors in **R**^{n}, then we may write

The vectors **e**_{i} 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 *i* ≠ *j*,

Thus in general we can say that:

Where δ _{ij} is the Kronecker delta.

Also, by the geometric definition, for any vector **e**_{i} and a vector **a**, we note

where *a*_{i} is the component of vector **a** in the direction of **e**_{i}.

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.

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

**Commutative:**- which follows from the definition (
*θ*is the angle between**a**and**b**):

**Distributive over vector addition:****Bilinear**:**Scalar multiplication:****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*(**a**⋅**b**) = (*c***a**) ⋅**b**=**a**⋅ (*c***b**).^{[7]}**Orthogonal:**- Two non-zero vectors
**a**and**b**are*orthogonal*if and only if**a**⋅**b**= 0.

- Two non-zero vectors
**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
**a**⋅**b**=**a**⋅**c**and**a**≠**0**, then we can write:**a**⋅ (**b**−**c**) = 0 by the distributive law; the result above says this just means that**a**is perpendicular to (**b**−**c**), which still allows (**b**−**c**) ≠**0**, and therefore**b**≠**c**.

- Unlike multiplication of ordinary numbers, where if
**Product Rule:**If**a**and**b**are functions, then the derivative (denoted by a prime ′) of**a**⋅**b**is**a**′ ⋅**b**+**a**⋅**b**′.

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

which is the law of cosines.

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.

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]}

- Mechanical work is the dot product of force and displacement vectors.
- Magnetic flux is the dot product of the magnetic field and the vector area.

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 *b** _{i}* is the complex conjugate of

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.

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.

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 ≤ *k* ≤ *n*}, and *u*_{i} 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 *a* ≤ *x* ≤ *b* (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]}

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

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.

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.

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.

A dot product function is included in BLAS level 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]}

- ^
^{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. - ^
^{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. **^**A I Borisenko; I E Taparov (1968).*Vector and tensor analysis with applications*. Translated by Richard Silverman. Dover. p. 14.**^**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..**^**Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DotProduct.html**^**T. Banchoff; J. Wermer (1983).*Linear Algebra Through Geometry*. Springer Science & Business Media. p. 12. ISBN 978-1-4684-0161-5.**^**A. Bedford; Wallace L. Fowler (2008).*Engineering Mechanics: Statics*(5th ed.). Prentice Hall. p. 60. ISBN 978-0-13-612915-8.**^**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.**^**M. Mansfield; C. O’Sullivan (2011).*Understanding Physics*(4th ed.). John Wiley & Sons. ISBN 978-0-47-0746370.

- Hazewinkel, Michiel, ed. (2001) [1994], "Inner product",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weisstein, Eric W. "Dot product".
*MathWorld*.

- Explanation of dot product including with complex vectors
- "Dot Product" by Bruce Torrence, Wolfram Demonstrations Project, 2007.

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