What is Dot product: Definition and 388 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. docnet

    Help me understand why this summation index is not j

    The below image is an excerpt from a website about Markov Chains. In the red boxed which I put in the image, I don't understand why the term ##g(i)## isn't being summed over ##j## instead of ##i##, since the outer sum is over the ##i##th element of the vector ##Pg##, which is the dot product...
  2. N

    B Confused about dot product multiplication

    I'm confused about what we are really measuring when taking the dot product of two vectors. When we say we are measure "how much one vector points in the direction of the other", that description is not clear. At first I thought it meant how much of a shadow one vector casts on another and I...
  3. A

    Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

    I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
  4. 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...
  5. nomadreid

    I Dot product, inner product, and projections

    In simple Euclidean space: From trig, we have , for u and v separated by angle Θ, the length of the projection of u onto v is |u|cosΘ; then from one definition of the dot product Θ=arcos(|u|⋅|v|/(u⋅v)); putting them together, I get the length of the projection of u onto v is u⋅v/|v|. Then I...
  6. billard

    I Deriving formula for kinetic energy

    Hello! I am new to the differential version of classical physics, and I am trying to work how to derive kinetic energy from some pre-assumed equations: Assume that we know: ##\ddot{z} = 0## and ##m\ddot{\textbf{r}} \cdot \dot{\textbf{r}} = 0##This results in $$\frac{1}{2}m\dot{r}^2 = W =...
  7. Skaiserollz89

    Symmetry of an Integral of a Dot product

    This homework statement comes from a research paper that was published in SPIE Optical Engineering. The integral $$\int\int_{-\infty}^{\infty}drdr'W(\vec{r})W(\vec{r'}) \vec{r} \cdot \vec{r'}=0$$ is an assumtion that is made via the following statement from the paper : "Since...
  8. C

    I Absolute value bars in dot product derivation

    Dose someone please know why they have the absolute value bars in this derivation? many thanks!
  9. TGV320

    I Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors

    HI, I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ Why is it like that? Thanks
  10. O

    I Dot product of two vector operators in unusual coordinates

    Hi. I hope everyone is well. I'm just an old person struggling to make sense of something I've read and I would be very grateful for some assistance. This is one of my first posts and I'm not sure all the LaTeX encoding is working, sorry. Your help pages suggested I add as much detail as...
  11. J

    Understanding the Dot Product and Cross Product in Vector Calculations

    Could anyone explain the reasoning from step 2 to step 3? Specifically, I don't understand how to find the product of a cross product and a vector - like (v1 · v2)v1 and (v1 · v3)v1. I'm also confused by v1 × v3 + (v1 · v3)v1 -- is v1 × v3 = v1v3? How would this be added to (v1 · v3)v1? Thank you.
  12. D

    Why does the dot product in this solution equal zero?

    Hi everyone I have the solutions for the problem. It makes sense except for one particular step. Why does the dot product of a and b equal zero? I thought this would only be the case if a and b were at right angles to each other. The solutions seem to be a general proof and should work for...
  13. H

    I Sum of the dot product of complex vectors

    Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...
  14. A

    How do I calculate the work done by a force field using the dot product?

    y = 10*(1 + cos(0.1*x)) --> dy/dx = -sin(0.1x) dW = F*dx + F*dy = 10*sin(0.1*x)dx + 10*sin(0.1*x)*-sin(0.1x) integrating we have -100*cos(0.1*x) -10*sin(0.1x)^2 from 0 to 10*pi = W = 43 J. The answer says 257 J. Where am I wrong here?
  15. A

    I Confused about dot product of a and b = |a||b| if theta = 0

    I am not sure what I am doing wrong but dot product of a and b =/= |a||b| when I am trying to calculate it. Theta = 0: dot product(a and b) = ax*bx + ay*by |a||b|= sqrt((ax^2+ay^2)*(ax^2 + by^2)) = sqrt((ax*bx)^2 + (ax*by)^2 + (ay*bx)^2 + (ay*by)^2) =/= ax*bx + ay*by What am I doing wrong?
  16. M

    Dot product: ##\vec{D} \cdot\vec{E}## in SI units

    I'm trying to calculate the electrostatic energy, and I'm wondering what happens when I dot the D-field and E-field, with Si-units V/m**2. This is my equation: D dot E = (-4x(epsilon) V/m**2)(-4x V/m**2) + (-12y(epsilon) V/m**2)(-12y V/m**2) Are the final Si-unit still V/m**2 or V**2/m**4?
  17. U

    Vectors in yz and xz plane dot product, cross product, and angle

    I tried to find the components of the vectors. ##a_y =2.60 sin 63.0 = 2.32## and assuming the z axis would behave the same as an x-axis ##a_z =2.60 cos 63.0 = 1.18## ##b_z =1.30 sin 51.0 = 1.01## making the same assumption ##b_x =1.3 cos 51.0 = 0.82## I now think I should have switched these...
  18. Blackbear38

    Using Inner Product Properties to Solve Vector Problems

    Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...
  19. B

    I Why is (N dot N) different for magnitude than for X, Y, Z components?

    I have to perform a calculation on my data. Here is an example of data from just one time step (data from other time steps would appear as additional rows). X Y Z Total 2 2 1 3 Total = SQRT(X2 + Y2 + Z2). The calculation I have to do is: (N • N), where "N" is an average. I tried...
  20. greg_rack

    Basic vector operations, using cross and dot product

    Hi guys, I am losing my mind over this passage... I cannot understand how to get from the first expression with the cross products to the second ##\dot{\textbf{r}}(\textbf{r}\cdot \textbf{r})-\textbf{r}(\textbf{r}\cdot\dot{\textbf{r}})##
  21. P

    I Proof concerning the Four Fundamental Spaces

    Hello all, I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the row space to the null space column space to the left null space ------------------------------------------------ In the book of G. Strang there is this nice picture...
  22. Another

    Problem about dot product in probability density problem

    I don't understand why ? ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅(A \Psi ^* \Psi) ## form ## ∇ ⋅ (fg) = ∇f ⋅ g + f(∇ ⋅ g) ## Attempt at a Solution ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅ (A \Psi ^* \Psi) - ∇\Psi ^* ⋅ A\Psi - ∇\Psi ⋅ (A\Psi ^*) ##
  23. F

    I Dot product in spherical coordinates

    I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...
  24. H

    Proof of a dot product using sigma notation

    Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...
  25. S

    I Why should a Fourier transform not be a change of basis?

    I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
  26. LCSphysicist

    Solve this vector system containing sum and dot product equations

    Seems to me the answer is a specific vector: The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!) How to write this in vector language?
  27. T

    I Dot product in Euclidean Space

    Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...
  28. D

    I Fundamentals of Astrodynamics: Dot Product Question

    I'm reading Fundamentals of Astordynamics by Bate, Mueller, White and having trouble with this passage (pg15): "2. Since in general a⋅a' = a a'..." I don't think that this is the case. For instance in uniform circular motion r⋅r' = 0. Would appreciate if anyone has some insight into this.
  29. P

    O and P whose 4-velocity and 4-acceleration have a dot product of 0

    If ##\tilde{U}_0 \cdot \tilde{A} = 0## in one frame then I would imagine it is also zero in another frame because from my understanding is that dot products are invariant under boosts. So let's boost to the rest frame of O. In that frame ##\tilde{U}_{0T} = \left( c, 0,0,0 \right)## and as...
  30. G

    Determining the power of frictional force

    I can say that the frictional force always against the rolling sphere and the velocity is increasing for the ball. So The dot product F.v keeps on getting more and more negative, so how can the Pf remain constant? Well the velocity increases along the incline and the force of gravity is down...
  31. T

    I Dot Product with Derivative

    Summary: The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector 'b'. Hello, I have the following Problem. The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector...
  32. Kirkkh

    B Why is sine not used for dot product?

    There’s a old 2012 post on here “Why sine is used for cross product and cosine for dot product?” —there are a lot of great answers (which is how I came about this forum). After reading over the replies, it occurred to me: really it’s just because cosine is the “start” of a unit circle. Which...
  33. P

    Stuck on a few Vector homework problems

    I'm stuck on a few Vector homework problems. I don't quite understand how to write vectors A+B and A-B for questions 1b and 2b. I tried starting with calculating the magnitude for vector A+B on question 1b and then followed by finding theta, but I'm not sure if that's what I'm supposed to do...
  34. F

    Python Invert a matrix from a 4D array : equivalence or difference with indexes

    I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
  35. S

    MHB Find Dot Product Between Vector CD & Vector K

    Hi! I'm given 2 points C(2;6) and D(0;10), a vector A with its components = (-3, 2). I'm asked to find the dot product between vector CD and an unknown vector K, knowing that K is perpendicular to A, same norm as A and with a negative x-component. I know that perpendicular means the dot...
  36. A

    B Calculating the dot Product of \nabla and Vector Identity

    From the vector identity ##\nabla •fA=f(\nabla • A)+A•\nabla f## where f is a scalar and A is a vector. Now if f is an operator acting on A how does this formula change?? Like ##\nabla •[(v•\nabla)v]## where v is a vector
  37. P

    Intuition behind dot product

    I know that a dot product of 2, 2 dimension vectors a, b = (ax * bx) + (ay * by) but it also is equal to a*bCos(θ) because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the...
  38. ibkev

    I Dot product definition: deriving component form

    ## \newcommand{\ihat}{\hat{\boldsymbol{\imath}}} \newcommand{\jhat}{\hat{\boldsymbol{\jmath}}} \newcommand{\khat}{\hat{\boldsymbol{k}}} ## Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...
  39. T

    I Strange Dot Product definition

    Hi i have seen in abook the dot product defined as follows: Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2] how this definition connect with the common one: Dot(A,B)=Sum(ai*bi) Thanks!
  40. L

    B Line Integral, Dot Product Confusion

    From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
  41. learning_physica

    Work: Dot Product and Integral?

    I’m having trouble understanding the relationship between how work is both a dot product and integral. I know that work equals F • D and also the integral of F(x): the area under the curve of F and D. However, let’s say that I have a force vector <3,4> and a displacement vector of <3,0>. The...
  42. archaic

    B Dot product scalar distributivity

    I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...
  43. M

    How can I prove that the dot product is distributive?

    Homework Statement Prove that for vectors A,B and C, A.(B+C)=A.B+A.C and prove the property that for two vectors, A and B the dot product is equal to A^ie_i . B^je_j = e_i.e_jA^iB^j Homework Equations Only use the definition where for two vectors a and b the (length of a)(length of b)cost...
  44. karush

    MHB 412.0.10 ok so going with dot product with 07312400508

    Use the UPC scheme to determine the check digit for the number $07312400508$. here is the example from the book ok so going with dot product with 07312400508 \begin{align*}\displaystyle &\quad (0731 2 4 0 0 5 0 8)\cdot(3,1,3,1,3,1,3,1,3)\\ &= 0\cdot3+7\cdot1 +3\cdot3+1\cdot1...
  45. M

    Dot product and basis vectors in a Euclidean Space

    Homework Statement I am asked to write an expression for the length of a vector V in terms of its dot product in an arbitrary system in Euclidean space. Homework EquationsThe Attempt at a Solution The dot product of a vector a with itself can be given by I a I2. Does that expression only apply...
  46. M

    I Simple dot product in polar coordinates

    Let's take two orthogonal curves in polar coordinates of the form ##\langle r,\theta \rangle##, say ##\langle r,0\rangle## and ##\langle r,\pi/2\rangle##. Cleary both lines are orthogonal, but the dot product is not zero. This must be since I do not have these vectors in the form ##\langle...
  47. maxknrd

    I More elegant way to solve divergence of arbitrary dotproduct

    This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
  48. thee qs

    Magnetic moment and magnetic field and dot product

    Homework Statement a magnetic moment of U = 1 (i) + 2 (k) , surrounded by a magnetic uniform field of B= 3 (i) + 4 (j) - 1 (k) find the potential energy in mJ ? Homework Equations dot product of 2 vectors ( ui*bi)+(uj*bj)+(uk*bk) = or finding the module of both vector and doing AB...
  49. T

    B Find Perpendicular Forces Given 20 N at 60 Degrees

    I have a question. If you're given a force of 20 N and its 60 degrees to the horizontal, how could you find two perpendicular forces?
  50. D

    I Differentiating vector dot product

    Hi. If I have a vector v , say for velocity for example then v.v = v2 and I differentiate wrt t v.v I get 2v.dv/dt but if I differentiate v2 I get 2v dv/dt but v.dv.dt is not the same as v dv/dt so what am I doing wrong ? Thanks