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. C

    Dot product in uniform circular motion question -- Finding angle?

    I've attached an image of part a of the question to this thread. My question is this (the solution to these former homework problems are posted to help us study for exam, which is why know this already): The angle between the two velocity vectors is determined to be pi/2. How? I know that dot...
  2. R

    Euclidean space: dot product and orthonormal basis

    Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...
  3. B

    Use vectors and the dot product to prove the midpoint

    Homework Statement [/B] Use vectors and the dot product to prove that the midpoint of the hypotenuse of a right triangle is equidistant to all three vertices. Homework Equations [/B] I know the dot product is A⋅B = |A||B|cosΘ ... or ... A1B1 + A2B2 + A3B3 ... + AnBn I know the...
  4. ognik

    Vector multiply that is NOT dot or cross?

    Hi - just working through my text (studying by correspondence) on Del operator - so Curl, div etc. Came across some identities parts of which which have me confused. what does it mean when a vector is shown as multiplying something - but without dot or cross? For example F(∇.G) or ∇(F.G) or...
  5. 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...
  6. A

    Finding a basis for a particular subspace with Dot Product restrictions

    Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3]. My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then, w-2x+3z=0 and 2x+y+3z=0 So, she solves the system and get the following: Subspace= {...
  7. S

    How can I calculate the speed of the kaon using four-momentum conservation?

    Homework Statement So a kaon moving at some speed in the +x direction spontaneously decays into one pion and one anti-pion. The anti-pion moves away with velocity of 0.8c, and the pion moves away with velocity of 0.9c. Mass of kaon = 498 MeV/c^2 Mass of pion/anti-pion = 140 MeV/c^2...
  8. J

    Understanding the Inner Product and Dot Product in Linear Algebra

    I'm just trying to understand from a linear algebra standpoint how they define dot product from the inner product and how this gives rise to a definition of length and angle. somehow there is a way to combine points in space to a scalar value that unambiguously determines length and angle? Is...
  9. M

    Confusion with Dot Product in Polar Coordinates with the Metric Tensor

    Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...
  10. A

    Can the Dot Product be Customized to Change Linearly?

    So, is there anyway to make the dot product change linearly? What I mean by this is when the angle is 45 degrees, I want it to be 0.5 instead of 0.7071 as you can see in this image: Instead I want 45 degrees to be 0.5, 60 degrees to be 0.33 and 30 degrees to be 0.66. Same would apply for...
  11. A

    Dot product of a vector and a derivative of that vector

    I'm reading through Douglas Gregory's Classical Mechanics, and at the start of chapter 6 he says that m \vec{v} \cdot \frac{d\vec{v}}{dt} = \frac{d}{dt}\left(\frac12 m \vec{v} \cdot \vec{v}\right), but I'm not sure how to get the right hand side from the left hand side. If someone could point...
  12. J

    Double dot product in Cylindrical Polar coordinates

    Hello, I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: 2W = σijεij Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: 2W = σxxεxx +...
  13. A

    Cross and dot product of two vectors in non-orthogonal coordinate

    Hi everyone, I have to find out how to do cross and dot product for two vectors in non-orthogonal coordinate system. thanks
  14. M

    Dot Product of Equilateral Triangle

    Homework Statement In an equilateral triangle with sides u, v, w, where they are all unit vectors, find u dot w. Homework Equations u dot w = |u||w|cosθ The Attempt at a Solution The answer is ##\frac {-1} {2} ## cos(120) = -1/2 Elsewhere, I read the statement that since these are...
  15. 22990atinesh

    Intuitive meaning of Dot Product

    I know intuitively that the Cross Product of two vectors ##\vec{A}## and ##\vec{B}## represents another vector ##\vec{A \times B}## perpendicular to it. In study of physics we come across this situation a lot. Hence I can visualize some applications of it I know that the dot product of...
  16. E

    Dot Product of a Unit Vector with the Negative of itself

    Homework Statement I am trying to calculate the flux for the octant of a sphere, and I am trying to figure out how the mathematics, dot products, and dA works in the integral. I already did the quadrant for \hat{θ} where θ= π/2 (the bottom quadrant) and I did the left quadrant where \hat{n}...
  17. O

    Vectors - dot product and cross product?

    Vectors -- dot product and cross product? Hello may i know when to dot product and cross product?? both look to same to me..
  18. E

    Can a dot product be negative in case of length?

    Let's say A and B are 2 vectors with length in cm and the angle between them is 170°. Obviously, the dot product of A and B will give cm2 as unit but since the value of cos(170) is negative, will the dot product be negative (something)cm2?
  19. P

    How Do Dot and Cross Products Differ in Describing Physical Phenomena?

    I am trying to understand the difference from a physical phenomena point of view, not just math. Surprisingly I think I got the cross product like in rotational momentum. You have the momentum vector and we have effective distance from the momentum vector R that needs to be perpendicular to the...
  20. M

    Dot Product of a Vector and its Derivative- Reality

    Hey everyone, This has been bugging me for a bit. I think I'm probably missing something pretty easy. A dot B= ABcos(θ), where θ is the angle between A and B. There is the little shortcut that says where B is the derivative of A, A dot B= AB. Clearly then cos(θ) = 1, and the angle between a...
  21. Lebombo

    Projection Using Dot Product Finding a Force (Boat Problem)

    "Projection Using Dot Product" "Finding a Force" (Boat Problem) Homework Statement ------------------------------------------------------------------------------------------- A 600 pound boat sits on a ramp inclined at 30 degrees. What force is required to keep the boat from rolling down...
  22. P

    Dot product of vector and del.

    I'm not sure which section is best to post this question in. I was wondering if the expression (u $ ∇) is the same as (∇ $ u). Here $ represents the dot product (I couldn't find this symbol. ∇=del, the vector differentiation operator and u is the velocity vector or any other vector
  23. S

    MHB These two problems are based on Vectors, dot product and distance for sphere.

    Problem 1: Let S1 be a sphere centered at(0, 1, -3) with radius 1 and let S2 be a sphere centered at (3, 5, -9) with radius 2. Find the distance between the two spheres. problem 2: Given three non-zero vectors v1, v2, v3 we say that they are mutually orthogonal when v1 dot v2= 0, v1 dot v3=0 ...
  24. T

    Find the True Statement About Dot Product of Two Vectors

    Homework Statement The dot product of.two vectors is -1which of the following statements is true A. They must be unit vectors pointing in opposite directions. B. They must be unit vectors pointing j. The same direction. C. They must be more than 90( and less than 270 )degrees from each...
  25. L

    Show that the dot product is linear: Bra-ket notation

    Homework Statement Show that the dot product in two-dimensional space is linear: <u|(|v> + |w>) = <u|v> + <u|w> The Attempt at a Solution I feel like I'm missing some grasp of the concept here ... I would think to just distribute the <u| and be done in that one step, but I'm being...
  26. S

    Prove that the absolute value of a dot product is less than or equal t

    Homework Statement That is prove that |a•c|≤|a||c| for any vector a=<a1,a2,a3> & c=<c1,c2,c3> Homework Equations The Attempt at a Solution I really don't have much of an attempt at the solution. I am not sure where to start. I can kind of justify it in my mind by saying the...
  27. T

    Calculating a theta using dot product in 3D coordinate

    I'm so confused about finding an angle, theta in this illustration. With having three coordinate information, how can I calculate the theta using dot product? I would easily find the angle by using trigonometric formula if I ignore the z-axis. But I want to solve this problem with...
  28. J

    Inner Product vs Dot Product: Understanding the Difference

    A simple question: what is the difference between inner product and dot product?
  29. F

    Can we simplify the integral of a dot product to just the product itself?

    Hello, I have a quick question about integrals of dot products. We are learning about magnetic flux as the integral of b dot da. However, what circumstances must be present where we can simplify this integral into (b*a) and ignore the integral?
  30. H

    Changing Dot Product to Simple Multiplication

    How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following: eb.\partialcea=-\Gammaa bc How do I get just an expression for \partialcea?
  31. J

    Proof of Dot Product Vector Equation: u•v = ||u|| ||v|| cos (theta)

    Homework Statement Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the...
  32. P

    Derive cross product from dot product

    can you show me derive cross product from dot product?
  33. W

    Gradient of the dot product of two vectors that are the same

    Hi, I am trying find the simplified expression of this: ∇(E \cdot E) Where E is the electric field that can written as E_{0}(exp(i(kx-ωt)) I know that since the two vectors are the same => E \cdot E = ||E||^{2} Do I take the gradient of the magnitude then? It just doesn't feel...
  34. PsychonautQQ

    Cross product and Dot product problem

    Homework Statement if v x w = <5,5,-2> (v cross w) and v * w = 6 (v dot w) then what is the tan(θ) between the two vectors v and w? The Attempt at a Solution well I was thinking v x w = |v||w|sinθ as well as v dot w (v*w) = |v|w|cosθ divide one equation by the other...
  35. PsychonautQQ

    Calculating Work Using Dot Product: Constant Force and Particle Position

    Homework Statement A constant force of 1i - 5j -8k moves (1,-4,2) (-3,2,-1), what is the work done on the particle? Homework Equations Avector*Bvector=ABsinθ ?? I think? The Attempt at a Solution I really am quite lost... but I found the coordinates for the position vector...
  36. G

    MHB Problem involving matrix multiplication and dot product in one proof

    The problem is: Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A). The way I went about starting this problem is to...
  37. D

    MHB How Do I Take the Dot Product of a Complex Expression with Itself?

    $$ \frac{(\dot{\mathbf{r}}\times\ddot{\mathbf{r}}) \times\dot{\mathbf{r}}}{\lvert\dot{\mathbf{r}} \rvert\lvert\dot{\mathbf{r}}\times\ddot{\mathbf{r}}\rvert} $$ How do I take that dot product of the expression of above with itself?
  38. D

    Cauchy Schwarz proof with alternative dot product definition

    Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...
  39. J

    Is A always equal to zero if it is perpendicular to every vector X?

    Homework Statement Let A be a vector perpendicular to every vector X. Show that A = O Edit: it is O not 0. (OH not zero) haHomework Equations So, we know if A and X are perpendicular then A(dot)X = 0 I see no reason why A would have to be equal to 0. Could X (not equal) 0? Could it be...
  40. S

    Trying to understand dot product of two DIFFERENT vectors

    Understanding the use of Pythagorean theorem for the length and square of a vector and how this is the dot product of a vector with itself is no problem. I'm trying to look inside the meaning of the dot product of two different vectors and understand it. I can also accept (just following the...
  41. V

    Dot Product Clarification (Kleppner & Kolenkow p.9)

    Problem: In Kleppner's book, Introduction to Mechanics, he states "By writing \vec{A} and \vec{B} as the sums of vectors along each of the coordinate axes, you can verify that \vec{A} \cdot \vec{B} = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}." He suggests summing vectors, but since the sum of two...
  42. O

    What is the Dot Product of Two 2x2 Matrices?

    This seems like a very basic question that I should know the answer to, but in my image processing class, my teacher explained that a basis set of images(matrices) are orthonormal. He said that the DOT product between two basis images (in this case two 2x2 matrices) is 0. so, for example...
  43. G

    What is the Error in Calculating the Integral of Dot Product?

    Hi all, I'm working on a math problem with a known answer - though I can't reproduce the maths. The problem is this: there is a random 3d vector of unit length with a uniform probability, \vec{v}, and a secondary unit vector \vec{u}. It is stated that: f = \int_{S^2}{| \vec{v} \cdot...
  44. cocopops12

    Is the Dot Product Definition Valid Only for Orthogonal Coordinates?

    The definition of the dot product is given by A = <a1,b1> B = <a2,b2> A dot B = a1a2 + b1b2 Is this definition valid for orthogonal coordinates only?
  45. N

    Orthogonal Vectors for Sphere Construction: Find Center and Radius

    R <x,y,z> A<a1,a2,a3> B<b1,b2,b3> Show that (r-a).(r-b)=0 represents a sphere find its center and radius So i see that if 2 vectors are orthogonal you can create a sphere and find the radius and center but can somone better explain this problem
  46. N

    Find 2 Unit Vectors at Angle π/3 with <3,4> using Dot Product

    find 2 unit vectors that make an angle of pi/3 with <3,4> <3,4>dot<a,b>=5/2=3a+4b b=5/8-3/4 a |<a,b>|=1 such that a^2+25/64+15/16a+9/16 a^2=1 25/16 a^2+15/16/ a=39/64 100a^2+60a=39 a^2+3/5a=39/100 (a+3/10)^2=48/100 a=(4sqrt(3)+-3)/10 so b=5/8-(12sqrt(3)+9)/40...
  47. N

    Find 2 Unit Vectors at 60 Degrees with <3,4> - Dot Product Calculation

    Find 2 unit vectors that make a 60 degree angle with <3,4> Vector <a,b> Taking b=1 cos60=1/2 3a+4=(5/2 )sqrt(a^2+1) 36a^2+96a+64=25a^2+25 11a^2+96a=-39 (sqrt(11)a+48)^2= 2265 +- a=(sqrt(2265)-48)/sqrt(11)
  48. E

    Is taking the square value of a dot product a valid mathematical operation?

    I was recently going through the proof of Compton scattering and I saw that they took a square value and wrote it as p^2=p(dot)p= etc... Is this true or all squared values?
  49. D

    Is the Dot Product of Two Vector Pairs Always Commutative?

    Homework Statement The Attempt at a Solution I am working a physics problem and want to make sure I'm not making a mistake in the math. Here is my math inquiry: Say you have (a*b)(c*d) where * indicates the dot product, and a,b,c, and d are all vectors. Can you say that (a*b)(c*d) =...
  50. E

    2 questions about dot product

    1. In the navier stokes equation we have the term (\vec{u} \bullet∇)\vec{u} If I have \vec{u} = f(r)(-y,x) with r= \sqrt{x^2+y^2} then is there some some of product rule/identity that needs to be invoked for the initial dot product? I would say this calculation is...
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