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. K

    Dot Product Involving Path of a Curve

    Homework Statement Let ##\gamma(t)## be a path describing a level curve of ##f : \mathbb{R}^2 \to \mathbb{R}##. Show, for all ##t##, that ##( \nabla f ) (\gamma(t))## is orthogonal to ##\gamma ' (t)##Homework Equations ##\gamma(t) = ((x(t), y(t))## ##\gamma ' (t) = F(\gamma(t))## ##F = \nabla f...
  2. V

    Solve Dot Product Riddle in 3 or 1 Guess

    I was recently posed a riddle that went like the following: There are two people. Person A picks three numbers from 0-99. Person B guesses which three numbers that person A has picked. Then, person A gives the dot product of his picked numbers with person B's guessed numbers. The question is...
  3. C

    Finding angle through cross and dot product

    By using both cross and dot products, the angle between 2 vectors can be found. But there is 1 question that I tried for countless times that the result of cross product and dot product are not the same. Here is the vectors that I am talking about A= 2i+3j=k B= -4i+2j-k The result by...
  4. C

    The reverse operation for the dot product?

    Okay. First off. This isn't homework. I have been working on a personal project and if I could get an equation for this it would really help me out. I'll try and explain the problem the best I can. What I need: An equation/method where I can specify an angle theta and a 3D line and it...
  5. L

    Gradient of a dot product identity proof?

    Gradient of a dot product identity proof? Homework Statement I have been given a E&M homework assignment to prove all the vector identities in the front cover of Griffith's E&M textbook. I have trouble proving: (1) ∇(A\bulletB) = A×(∇×B)+B×(∇×A)+(A\bullet∇)B+(B\bullet∇)A Homework...
  6. A

    A Dot product analysis proof (might be simple)

    Homework Statement so we have a (D) Line (geometry) it's Cartesian equation is ax+by+c=0 we have an A(\alpha,\beta) prove that the distance between the line(D) and the point A is d=\frac{la\alpha+b\beta+cl}{\sqrt{a^2+b^2}}Homework Equations The Attempt at a Solution let every distance be a...
  7. D

    MHB Derivative dot product cross product

    $$ \frac{d}{dt}[\mathbf{a}\cdot (\mathbf{v}\times\mathbf{r})] = \dot{\mathbf{a}}\cdot (\mathbf{v}\times\mathbf{r}). $$ How is this true? Shouldn't the derivative affect the cross product as well?
  8. C

    Shedding some light on the dot product

    The dot product A . B is the magnitude of vector A times the projection of B onto A. B . A is the magnitude of vector B times the projection of A onto B. Correct? A . B = B . A and this makes sense. But, say you're trying to find the components of a vector V in the direction of a vector W...
  9. C

    Exploring the Dot Product: Arithmetic and Magnitude of Vectors

    If you square the magnitude of a vector you get the dot product, correct? ||v||^2 = v . v Can you also say that ||v|| = sqrt(v . v)?
  10. L

    Dot product, is (a.b).(a.b)=(a.a).(b.b) ?

    Hi, As I remember, dot product is commutative, and so (a.b).(a.b) = (a.a).(b.b) But when I apply to simple vectors it is all wrong, e.g: a = (2, 2, 0) b = (1, 0, 0) (a.b).(a.b) = 2.2 = 4 (a.a).(b.b) = 8.1 = 8 Why are they different? Pls explain for me Thanks
  11. A

    Deriving the algebraic definition the dot product

    Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
  12. A

    Proving the dot product between these vectors is always the same value

    How would you show that the dot product between the normal unit vector of a plane and a position vector to any point on the plane is always the same without using this formula n.(r-r_0) = 0 ∴ n.r=n.r_0 where n is the normal vector, r and r_o are two position vectors to two points on the...
  13. E

    Dot Product Proof: Prove |a|^2 = 2|b|^2

    Homework Statement The point P and Q have postion vectors a + b, and 3a - 2b respectively, relative to the origin O.Given that OPQR is a parallelogram express the vector PQ and PR in terms of a and b. By evaluating two scalar products show that if OPQR is a square then |a |2 = 2 |b |2 The...
  14. U

    Vector field dot product integration

    Homework Statement Calculate F=∇V, where V(x,y,z)= xye^z, and computer ∫F"dot"ds, where A)C is any curve from (1,1,0) to (3,e-1) B)C is a the boundary of the square 0≤x≤1, 0≤y≤1... oriented counterclockwise. Homework Equations ∫F"dot"ds= ∫F(c(t)"dot"c'(t) The Attempt at a Solution...
  15. F

    Vector geometry: Proof of a trapezium and cross and dot product help

    Homework Statement #1 Given that the angle between the vectors a and b is 2Pi/3 and |a|=3 and |b|=4 calculate: (axb)^2 [(2a+b)x(a+2b)]^2 #2 Given three unit vectors, a, b, c such that a+b+c=0 find (a dot b) + (b dot c) + (c dot a) #3 Given AB=a+2b BC=-4a-b CD= -5a-3b...
  16. M

    Dot Product Projection: What Does A Dot B Mean?

    Simple question, but I don't know why I never learned this before. If the scalar projection of vector B onto vector A is B * Unit vector of A (or [A dot B]/[magnitude of A]), then what does the dot product of simply A and B give you, assuming neither is a unit vector. If it's not clear what...
  17. F

    Dot product between arrays: basis representation of an image

    Hello Forum, When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors: x= a1 v1 + a2 v2 + a3 v3+ ... Each coefficient a_i is the dot product between x and each basis vector v_i. If the vector x is not a row (or column vector)...
  18. B

    Dot product = Product of norms

    Hey guys I am a beginner in linear algebra. I am doing vectors now and I just noticed that when two vectors are parallel (or antiparallel), the product of their norms is equal to the absolute value of their dot product, or |u \cdot v | = ||u|| \ ||v|| I know that this is a special case of...
  19. A

    Multiplication using dot product.

    Homework Statement Homework Equations Tan θ=A.B/|A|*|B| The Attempt at a Solution A=4i-9j B=9i-6j A.B=-18 Magnitude of A = √97 Magnitude of B = √117 Solve that out using Cos-1 (-18/Sqr97*Sqr117) and I keep getting the wrong answer.
  20. D

    Proof of Gradient Dot Product Identity

    Hey guys, this is for my classical E&M class but it's more of a math problem. Homework Statement Show: ∇(\vec{A} . \vec{B}) = \vec{B} \times (∇ \times \vec{A}) + (\vec{B} \times ∇)\vec{A} + \vec{A} \times (∇ \times \vec{B}) + (\vec{A} \times ∇)\vec{B} Homework Equations I tried...
  21. R

    Spherical coordinates, vector field and dot product

    Homework Statement Show that the vector fields A = ar(sin2θ)/r2+2aθ(sinθ)/r2 and B = rcosθar+raθ are everywhere parallel to each other. Homework Equations \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0) The Attempt at a Solution So, if the dot product equals 1. They should be...
  22. H

    Dot product of vector and a dyadic.

    Question If a = i - 2j + 2k b = 4i - 3j M = 3ii + 2ij - ji + kj. M is a dyadic. Determine a\bullet2M\bulletb Homework Equations The Attempt at a Solution This isn't really homework, but rather just some self study to help me understand some of my work better. Attached is the...
  23. putongren

    Dot product of acceleration and velocity

    Just wondering... If the dot product of the acceleration and velocity vectors is zero, then does v2/r = 0 have to be true? If this is true, is it possible to prove it? If the statement is false, is it possible to prove that as well?
  24. A

    Why Does the Projection of Vector A on B Not Equal Zero?

    greetings, consider two vector as it is A(1,0) and B(-1,0). now if we find the projection of A on B we should get zero but its coming -1.where i am going wrong? advanced thanks.
  25. matqkks

    MHB Exploring Backface Culling in Java: Utilizing Dot and Cross Products

    There is mention that dot products of vectors are used in backface removal in 3d graphics. Does anyone know of any article which explains the application of dot products to this area of graphics? I am looking for a concrete example.
  26. R

    Understanding Complex Numbers in Dot Product Calculations

    Homework Statement The Attempt at a Solution do you see where it sees sqrt((1-i)(1+i)+9)? It should be (1+i)(1+i) Why isn't it?
  27. H

    Angle between 2 vectors given dot product and cross product

    Homework Statement A dot B=0.707m^2, A cross B=4.950m^2 k^. If |A|=2.500m and B makes an angle of 135° with the positive x-axis, what are A and B in component form? Homework Equations A*B = |A||B|cos(θ) A X B = |A||B|sin(θ)@RHR The Attempt at a Solution I have no prior physics...
  28. D

    Why sine is used for cross product and cosine for dot product?

    While we calculate cross product of two vectors let A and B we write ABsinθ. And while we calculate dot product of them we write ABcosθ. Why particularly we use sinθ for cross product and cosθ for dot product.Is there any physical reason why we choose sine for cross product and cosine for...
  29. P

    What is the difference between dot product and cross product in physics?

    Why is dot product given by a*b cosθ whereas cross product ab sinθ.
  30. J

    Matrix multiplication vs dot product

    What is the difference between matrix multiplication and the dot product of two matrices? Is there a difference? If, A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} e & f \\ g & h \end{pmatrix} then does {\mathbf{A} \cdot...
  31. A

    Whats difference between inner product and dot product?

    i m really confused..please explain with a physical example so that I can learn the application of it
  32. M

    Applying the Product Rule to Vector Dot Products

    Homework Statement I have vector R. I need to show the R dot dR/dt = 0 => 1/2 d/dt[R dot R] Homework Equations The Attempt at a Solution I guess I've never really applied the chain rule to dot products and its throwing me off. How does one go from R.dR/dt=0 to 1/2 d/dt[R.R] = 0. I...
  33. Telemachus

    What is the Inner Product between Tensors in Cartesian System?

    Hi there. I have this problem, which says: In the cartesian system the tensor T, twice covariant has as components the elements of the matrix: \begin{bmatrix}{1}&{0}&{2}\\{3}&{4}&{1}\\{1}&{3}&{4}\end{bmatrix} If A=e_1+2e_2+3e_3 find the inner product between both tensors. Indicate the type and...
  34. S

    Dot product of Force and Position as a constant of motion - physical significance?

    Reason I posted this in the maths help forum is that an equation of this form randomly popped up in a homework I was doing on differential geometry. I started with a one-form ω=dβ (β is a scalar function) and found that if for a random vector v, ω(v) = 0, then \frac{d}{dt} \left(...
  35. H

    Dot product of a vector with the derivative of its unit vector

    Homework Statement Let c(t) be a path of class C1. Suppose that ||c(t)||>0 for all t. Show that c(t) dot product with d/dt((c(t))/||c(t)||) =0 for every t. Homework Equations I am having trouble with the derivative of (c(t)/||c(t)||) and how to show that when its dotted with c(t) that...
  36. S

    Please give me a hint to solving this simple vector dot product proof

    Homework Statement Let 'u' and 'v' be two non zero vectors such that the prjection of 'u' along 'v' equals the projection of 'v' along 'u.' Using the formula for projection, show that 'u' and 'v' are either perpendicular or parallel. Homework Equations The Attempt at a Solution...
  37. S

    Determinant as Dot Product in R^2 Question

    Homework Statement In R^2, vectors x = (x1, x2) and a = (a1, a2). For fixed a, det(a, x) is a scalar-valued linear function of the vector x. Thus it can be written as the dot product of x with some fixed vector w. Explain why w is perpendicular to a. Do not use an expression of w in terms of...
  38. B

    Triangle and dot product to find angle

    Vector A and vector B are expressed in component form. A = [2.32,-5.16,7.88] B = [-1.12,3.45,-12.8] The standard arrow representation of these vectors and that of can be arranged to form a triangle in a plane that represents the geometric equivalent of the subtraction operation. The...
  39. A

    Vectors A,B,C: Is Parallel Necessary?

    say for example for 2 vectors A.B=0 and for another pair A.C=0.is it necessary that B is parallel to C.If yes how?
  40. T

    Confused about the True Geometric Meaning of a Dot Product Answer.

    I have performed numerous calculations of dot products throughout my math courses, but I think I lack a fundamental understanding of what it actually means, beyond the abstract way I have been taught to deal with them. I know the definitions (it's the inner product, or the projection of A on to...
  41. L

    Dot Product confusion (no calculations involved)

    If I take the Dot Product of two vectors, say A and B, I get: AxBx + AyBy + AzBz And then when I add those terms, I get the magnitude, right? So when one of those terms are negative, that means I could end up with a negative magnitude? I thought magnitude had to be positive.
  42. L

    Dot product (scalar product) of 2 vectors: ABcos[itex]\theta[/itex]

    How, precisely, do you get/derive the Bcosθ term? Is it simply [Cosθ=A/B] --> [BCosθ = A] ? It can't be that simple because then how is the extra length of vector A fit into [*A*Bcosθ]? I feel pretty confused as to what is going on here. To summerize, A x B = [ABcosθ] makes little...
  43. U

    Ev(Unit Vector) and projection of a vector in a dot product

    So my book says Lets suppose, We have two vector v and u w=projection of u ev= unit vector θ=angle between the two w=(u.ev)ev or w=( (u.v)/(v.v) )v Now, the second equation is fairly easy to understand if we understand the first one because ev= v / |v| What is...
  44. M

    Dot product between two unit vectors

    Hello, I am trying to find the angle between two unit vectors but I was wondering what I am supposed to do when the dot product is greater than 1 or less than -1. For example -0.0288067i + -0.989524j + -0.141463k 0.169194i + -0.0644865j + -0.983471k
  45. J

    Deriving cross product and dot product, stuck at beginning.

    Homework Statement Assuming that ∅ is a differentiable scalar valued function and F a differentiable vector field, derive the following identities. a)∇(dotted with)(∅F) = ∇∅(dotted with)F + ∅∇(dotted with)F b)∇(crossed with)(∅F) = ∇∅(crossed with)F + ∅∇(crossed with)F Homework Equations The...
  46. A

    Simple vs. Dot Product: Exploring the Differences

    whats the difference between simple product and dot product?
  47. F

    Dot product geometric proof question?

    Dot product proof question? Hi, I'm having trouble understanding the proof of the dot product in three dimensions (not using the cosine rule approach). Here's what I have for the 2D proof: u = u1 i + u2 j v = v1 i + v2 j u.v = u1v1 + u2v2 u.v = |u| |v| cos(θ) => u1v1 + u2v2 = |u| |v|...
  48. H

    [Dot Product] Vector Proection

    [Dot Product] Vector Projection Homework Statement Homework Equations The Attempt at a Solution I am not sure what to do here -- I know that the projection of u onto a "dotted" with w = 0 by definition, but I don't know how to show this. added this second part after plugging in...
  49. B

    What is the Dot Product of a Cutting Tool Under Microprocessor Control?

    Homework Statement A cutting tool under microprocessor control has several forces acting on it. One force is \vec{F}=-αxy2\hat{j}, a force in the negative y-direction whose magnitude depends on the position of the tool. The constant is α=2.50 N/m3. Consider the displacement of the tool from...
  50. M

    Linear Algebra: Dot Product and Orthogonality

    This is from my homework, I was moving along nicely until I hit this problem, (there's another just like it right after this). I can't find reference for solving this in the chapter I am looking at. The answer is in the back of the book….-2911. Can someone explain this to me? ||\mathbf{u}|| =...
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