What is Scalar product: Definition and 91 Discussions

In mathematics, the dot product or scalar product 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. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
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, as is the case for the vector product in three-dimensional space.

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  1. cianfa72

    I Inner product vs dot/scalar product

    Hi, from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things. Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
  2. L

    What is the formula for the norm of a vector cross product?

    Hi everyone, I'm having problems with task c In the task, the norm has already been defined, i.e. ##||\vec{c}||=\sqrt{\langle \vec{c}, \vec{c} \rangle }## I therefore first wanted to calculate the scalar product of the cross product, i.e. ##\langle \vec{a} \times \vec{b} , \vec{a} \times...
  3. L

    Exploring the Role of Determinants in Orienting Scientific Research

    Hi, unfortunately, I have problems with the task c I used the tip with the Laplace evolution theorem and rewrote the determinant to calculate ##W_n##. Then I simply formed the scalar product ##W_1 W_n## and here I get now no further.
  4. P

    I Question about implication from scalar product

    Hi, Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have...
  5. D

    I Scalar product and generalised coordinates

    Hi If i have 2 general vectors written in Cartesian coordinates then the scalar product a.b can be written as aibi because the basis vectors are an orthonormal basis. In Hamiltonian mechanics i have seen the Hamiltonian written as H = pivi - L where L is the lagrangian and v is the time...
  6. twilder

    I Scalar product of biharmonic friction with velocity components

    I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is $$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2 $$ where ##\underline u = (u,v)## and ##A## is a...
  7. andylatham82

    B What does the scalar product of two displacements represent?

    Hi, This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...
  8. J

    Generalized coordinates- scalar product

    Homework Statement a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself. b: What would be the r, θ components of the unit vector in the θ direction? Homework Equations Scalar product of 2 vectors = AαgαβBβ The Attempt at a Solution For part a, I used the...
  9. T

    Potential of particle - why is there a scalar product here?

    I'm reading up on the Lagrangian equation, but what I'm asking is to do with electromagnetism. In the first equation here: http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf L equals the kinetic minus the potential energy. For the potential energy term, I just don't...
  10. F

    I Demo of cosine direction with curvilinear coordinates

    1) Firstly, in the context of a dot product with Einstein notation : $$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$ with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
  11. P

    B Minkowski metric, scalar product, why the minus sign?

    In Schutz's A First Course in General Relativity (second edition, page 45, in the context of special relativity) he gives the scalar product of four basis vectors in a frame as follows: $$\vec{e}_{0}\cdot\vec{e}_{0}=-1,$$...
  12. P

    Show this integral defines a scalar product.

    Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
  13. hugo_faurand

    B What is the reality of the scalar product?

    Hi everyone ! I would like to know the real meaning of scalar product. So, I know scalar product is defined as : ||a||.||b||.cos(a;b) = k But what k is ? (Sorry for my english, I am french). Regards :)
  14. G

    I Solve Scalar Product Problem in Set R of Functions [0,1]

    Alright, so we ran into a peculiarity in answering this question. Let R be the set of all functions f defined on the interval [0,1] such that - (1) f(t) is nonzero at no more than countably many points t1, t2, . . . (2) Σi = 1 to ∞ f2(ti) < ∞ . Define addition of elements and multiplication...
  15. L

    A Gradient of scalar product

    It is very well known result that ##grad[e^{i\vec{k}\cdot \vec{r}}]=i\vec{k}e^{i\vec{k}\cdot \vec{r}}##. Also ##\vec{k}\cdot \vec{r}=kr\cos \theta## and ##gradf(r)=\frac{df}{dr} grad r##. Then I can write grad e^{ikr\cos \theta}=ik\cos \theta e^{i \vec{k}\cdot \vec{r}}...
  16. andylatham82

    I How to find angle between vectors from dot and cross product

    Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy
  17. andylatham82

    I Angle between vectors via scalar product vs vector product

    Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...
  18. P

    I A directional, partial derivative of a scalar product?

    Let's say I have two vector fields a(x,y,z) and b(x,y,z). Let's say I have a scalar field f equal to a•b. I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b. Ideally, I would like to be able to express...
  19. R

    MHB Proving Positive Definite Scalar Product for $n \times n$ Matrices

    Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product: $$\langle X, Y \rangle = tr(X Y^T),$$ and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are (1) $\langle X, X\rangle \geq 0$ and (2)...
  20. ATY

    I Complex conjugation in scalar product?

    Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...
  21. S

    I Questions about gradient and scalar product

    I recently learned that the general formula for the dot product between two vectors A and B is: gμνAμBν Well, I now have a few questions: 1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...
  22. D

    I Exploring the Properties of Scalar Product & Law of Cosines

    Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##. With the construct of a triangle, the Law of Cosines is proved. ##\mathbf A## points to the tail of ##\mathbf B##. Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...
  23. prashant singh

    I Why does A.A = ||A||^2 in the scalar product formula?

    Why A.A = ||A||^2 , I know that from product rule we can prove this where theta =0 , I am asking this because I have seen many proves for A.B = ||A||||B||cos(theta) and to prove this they have used A.A = ||A||^2, how can they use this , this is the result of dot product formula. I havee seen...
  24. prashant singh

    I Scalar product and vector product

    why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove...
  25. A

    Bessel functions and the dirac delta

    Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
  26. S

    Why work done by a force is a scalar product

    Why work done by a force was taken as dot product between force applied and displacement caused?
  27. G

    Scalar product using right hand rule ?

    Homework Statement Refer to solution II , the author used the scalar analysis( dot product) to get the direction of moment ...IMO , this is incorrect ... Only cross product can be determined this way . correct me if I'm wrong . Homework EquationsThe Attempt at a Solution
  28. T

    Prove Determinant Using the Triple Scalar Product

    Homework Statement I'm supposed to prove det A = \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{kr} using the triple scalar product. Homework Equations \frac{1}{6} \epsilon_{ijk} \epsilon_{pqr} A_{ip} A_{jq} A_{ kr} (\vec u \times \vec v) \cdot \vec w = u_i v_j w_k...
  29. K

    Scalar product and the Kronecker delta symbol

    From a textbook. proof that the scalar product ##A\centerdot B## is a scalar: Vectors A' and B' are formed by rotating vectors A and B: $$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$ $$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...
  30. K

    Prove the scalar product

    Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS: ##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'## I posted this question long ago and i was told that this is the scalar product...
  31. Tony Stark

    Scalar Product of Orthonormal Basis: Equal to 1?

    What is the scalar product of orthonormal basis? is it equal to 1 why is a.b=ηαβaαbβ having dissimilar value
  32. Tony Stark

    Scalar Product of displacement four vector

    Homework Statement How does the scalar product of displacement four vector with itself give the square of the distance between them? Homework Equations (Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector) or how ds2=ηαβdxαdxβ The Attempt at a Solution Clearly I am completely new to the...
  33. E

    Calculating Power of a Solar Panel

    Homework Statement If at some particular place and time the sun light is incident on the surface of the Earth along a direction defined by the unitary vector – vˆ , where vˆ =(4, 3, 5)/sqrt (50) and with a power density P, what is the total power captured by a solar panel of 1.4 m2 and with...
  34. L

    Cosine question. Scalar product.

    Homework Statement Find angle between vectors if \cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...
  35. S

    Find Value of α for Scalar Product a\cdotb = 0 & Explain Phys. Significance

    Vectors a and b correspond to the vectors from the origin to the points A with co-ordinates (3,4,0) and B with co-ordinates (α,4, 2) respectively. Find a value of α that makes the scalar product a\cdotb equal to zero, and explain the physical significance. My attempt: The scalar product...
  36. D

    Is it a scalar product? I'm kind of lost

    The Vector A points 17° counterclockwise from the positive x axis. Vector B lues in the first cuadrant of the xy plane. The magnitudes of the cross product and the dot product are the same: i.e, |AXB|= |A(times)B| What Angle does B make with the positive x axis? 2. Is ti a scalar...
  37. H

    Math Methods: help with scalar product properties.

    Homework Statement For what values of k is (scalar product of vectors a and b) = a_{1}b_{1}-a_{1}b_{2}-a_{2}b_{1}+ka_{2}b_{2} a valid scalar product? The vectors a and b are defined as: a = a_{1}e_{1} + a_{2}e_{2} b = b_{1}e_{1} + b_{2}e_{2} where e_{1} and e_{2} are unit vectors...
  38. F

    MHB Defining Real-Valued Scalar Product in Vector Spaces

    Hi, can somebody help me with the problem: Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows...
  39. P

    Scalar product used for length?

    I got asked how the scalar product can be used to find the length of a vector? Could someone please explain
  40. F

    Tensor Notation for Triple Scalar Product Squared

    Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...
  41. M

    Is the Triple Scalar Product Always Zero?

    Hello, I am confused how vectors that are coplanar will give a triple product of zero? Or is it the case that all 3 vectors must be coplanar for a triple product of zero, or is 2 sufficient? I.e. the vector being dotted with one of the vectors being crossed in the same plane, will this...
  42. U

    Trace of Matrix Product as Scalar Product

    Homework Statement Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A) Show that this indeed fulfils the requirements on a scalar product. Homework Equations Conditions for a scalar...
  43. U

    Prove scalar product of square-integrable functions

    Homework Statement Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product? Homework Equations The Attempt at a...
  44. P

    Understanding Triple Scalar Product and Its Properties: Explained Simply

    Im having trouble understanding this property my book states that: a.(bxc) = b.(cxa) = c.(axb) it also states that a.(ax(anything)) = 0 I understand the second point and why that's true, what I don't understand is why a.(bxc) = b.(cxa) = c.(axb) is true If I name any 3 vectors a b...
  45. S

    Physics: Vectors & their scalar product

    Homework Statement Given the vectors: P = 8i +5j-Pzk m and Q = 3i -4j-2k m Determine the value of Pz so that the scalar product of the two vectors will be 60m2Homework Equations Sure seems like we will need to use the following equation: P * Q = |P| * |Q| * cos ∅ But I don't recall being able...
  46. G

    Why is scalar product of momentum in electron scattering conserved?

    Homework Statement Reading a textbook, I come across a situation where an electron is scattered off a nucleus. The book says p.P = p'.P', where p is the momentum of the electron and P is the momentum of the nucleus. I don't understand how it gets the conservation of scalar product. It's steps...
  47. K

    The scalar product of 4-vectors in special relativity

    Homework Statement I'm confused about the difference between the following two statements: \mathbf{V_1}\mathbf{V_2}=V_1V_2\cosh (\phi) and \mathbf{V_1}\mathbf{V_2}=\gamma c^2 Where \gamma is the Lorentz factor of the relative speed between the two vectors. Both vectors are time-like. The...
  48. B

    Scalar product square matrix hermitian adjoint proof

    Homework Statement If M is a square matrix, prove: (A, MB) = (adj(M)A, B) where (A, MB) denotes the scalar product of the matrices and adj() is the adjoint (hermitian adjoint, transpose of complex conjugate, M-dagger, whatever you want to call it!) Homework Equations adj(M)=M(transpose of...
  49. B

    Scalar product in spherical coordinates

    Hello! I seem to have a problem with spherical coordinates (they don't like me sadly) and I will try to explain it here. I need to calculate a scalar product of two vectors \vec{x},\vec{y} from real 3d Euclidean space. If we make the standard coordinate change to spherical coordinates we can...
  50. E

    Finding volume using the triple scalar product (vector calculus))

    Of the 3 vectors, does it matter what order I cross / dot them? <a \times b> \bullet c =? <a \times c> \bullet b