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.

View More On Wikipedia.org
Recent contents Top users
  1. B

    Dot product in spherical coordinates

    Homework Statement What is the dot product of two unit vectors in spherical coordinates?Homework Equations A∙B = ||A|| ||B|| cos(\theta) = cos(\theta)The Attempt at a Solution The above equation is the only relevant form of the dot product in terms of the angle \theta that I can find. However...
  2. M

    Surface Integral: dot product of two unit vectors

    Hello. :smile: I understand most of the work involved with these types of questions, but there is one point in an example I'm following that I don't understand. Homework Statement Evaluate: I = \int{(z^2)}dS over the positive quadrant of a sphere, where (x,y > 0). Homework...
  3. I

    How to Calculate Dot Product of Two Vectors?

    Homework Statement obtain the dot product of the two vectors F= 10i + 6j - 3k lb and B= 6i -2j ft Homework Equations F dot B = FxBx + FyBy + FzBz x is for the x components y is for the y components z is for the z components The Attempt at a Solution F dot B = (60 + -12 + 0) I ended...
  4. N

    Matrix Dot Product: Solving Linear Algebra

    Hi, here's the problem: for m = {{a, b}, {c, d}}, m \cdot m is suppose to = {{a^2 + b c, a b + b d}, {a c + c d, b c + d^2}} It's been ages since I took linear algebra and now can't figure out how this works. Thanks for your help!
  5. N

    Integrating a dot product inside an exponential

    Homework Statement This is from Peskin & Schroeder p. 14 in case anybody's interested. The function is U(t)=\frac{1}{(2\pi)^3}\int d^3p\, e^{-it \sqrt{p^2+m^2}}e^{i\vec p\cdot(\vec x-\vec x_0)} Homework Equations The Attempt at a Solution Essentially you write out the dot product as p\cdot...
  6. O

    Help Understanding Dot Product

    I'm learning about dot products, and I'm having a bit of trouble grasping why axbx + ayby + azbz = ab*cosΘ. I understand how it works in two dimensions, I think, but three is still fuzzy. This is what I came up with for two dimensions. The angle between the vectors is simply the difference...
  7. D

    What is the angle between B and C if A.B = 0 and A x C = 0?

    Homework Statement Given A . B=0 and A x C=0...whats the angle b/w B and C? Homework Equations The Attempt at a Solution..
  8. Y

    Question on dot product of vectors.

    \vec r = \hat x x + \hat y y + \hat z z \;\Rightarrow \;r = \sqrt { x^2+y^2+z^2} \;,\, \hat r= \frac { \hat x x + \hat y y + \hat z z}{ r} I want to find the dot product (\hat x -\hat r \frac x r) \cdot (\hat x -\hat r \frac x r) 1) \hat x -\hat r \frac x r = \hat x - \frac { (\hat x x...
  9. 2

    Integral of an integral of a dot product?

    Homework Statement A coil with N loops and radius R is surrounds a very long solenoid of radius r and n turns per meter. The current in the solenoid is varying sinusoidally with time according to the relation I(t)=I0sin(2πft) where I0 is the maximum value of the current, and f is its...
  10. P

    Does dot product always commute?

    I thought the dot product was commutative but there must be something about it that I don't understand. Perhaps the dot product is commutative only for vectors and not for tensors generally? In Kusse and Westwig, p70, it says that the order of terms matters because, in general, \hat{e}_j...
  11. B

    Calculate Torque Vector for Force F at r - Dot Product

    Homework Statement A force F=3i ̂+0.5j ̂-5k ̂(N)is applied to a partial located at r=2i ̂+4k ̂(m) Determine the torque vector? Homework Equations The Attempt at a Solution
  12. B

    Dot product issue. system of equations

    Homework Statement As illustrated, a ball of mass m_1=0.25 kg and velocity V_(0_1=+5.00 m/s) collides head on with a ball of mass m_2=0.8 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after they collide...
  13. L

    Dot product in non-orthogonal basis system

    Hi again, I don't want it to seem like I'm spamming topics here, but I was hoping I could get help with this dillema, too. So, let's say that, in affine 2-dimensional space, we have some two, non-orthogonal, independent vectors, and we also pick some point for an origin O. This clearly...
  14. Y

    Verify derivative of a dot product.

    Let \vec w(t) \;,\; \vec v (t) be 3 space vectors that is a function of time t. I want to verify that: \frac {d(\vec w \cdot \vec v)}{dt} = \vec v \cdot \frac { d\vec w}{dt} \;+\; \vec w \cdot \frac { d\vec v}{dt} I work through the verification by splitting w and v into x, y, z...
  15. L

    Question about cross and dot product

    Homework Statement Note: \nu is del could not find it... I need to prove the \nu*E=0 and \nu x E=-dB/dtHomework Equations E(s\phizt)=(Acos(Kz=wt)/s)s^ The Attempt at a Solution so for the first one \nu*E=0 I thought it would be d/ds(E) but what they did was 1/s d/ds(s E) no idea why they...
  16. V

    How do i calculate this dot product

    Homework Statement i dot 1/√2( i + j) Homework Equations The Attempt at a Solution I think that perpendicular vectors are zero are dotted together, and parallel vectors dotted are 1. I am tempted do do the distributive property where i dot i is 1 and i dot j is zero, but I...
  17. N

    So, the property is not applicable to this problem.

    Homework Statement \vec r = <x, y, z>, r = \left | \vec r \right | The problem is, verify the identity: \nabla \cdot (r\vec r) = 4r Homework Equations My book has the following property: (c\vec a)\cdot \vec b = c(\vec a \cdot \vec b) The Attempt at a Solution I tried using...
  18. A

    Dot Product of a Vector with its Derivative

    Hi Guys, I am doing an Electromagnetism course at uni and we just derived Poynting's Theorem in class. However, he left steps for us to fill in and that is why I have a question. In the derivation we get the dot product: B dot dB/dt and E dot dE/dt where both B and E are...
  19. T

    Difference between Inner Product and Dot Product

    Could someone explain me the difference between the inner product and the dot product? Thanks all
  20. P

    Prove c Bisects Angle Between a & b | Dot Product Proof

    Homework Statement If c = |a|b + |b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b. Homework Equations cos\theta = (a.b)/(|a||b|) The Attempt at a Solution no idea. i struggle with proofs :( please help!
  21. T

    Proving Geometric Fact: u+v Perpendicular to u-v Using Dot Product

    "If u and v are any two vectors of the same length, use the dot product to show that u + v is perpendicular to u − v. What fact from geometry is does this represent." This is basically the last question in an assignment on vectors (first year university, linear algebra). The questions all focus...
  22. H

    Integration of the dot product of two vectors

    Hi, I am having problems understanding the difference between the equations in the files attached profile2.png. where W_i is just representing the ith component of the variable W, and n and v under bars are two vectors, and gamma is a boundary and gamma_e shows is a boundary. what...
  23. B

    Mathematica Mathematica Loop with Dot Product

    Basically, I'm trying to get the various entries of TT to be dotted with the various entries of KK. Each row of TT is a vector while in KK the elements are vectors. So I'm working with different dimensions, but should be able to "call" them still using the right command. My attempt is below...
  24. P

    Does the Dot Product of Two Non-Zero Vectors Bisect the Angle Between Them?

    Homework Statement if c = |a|b + |b|a where a b and c are all non zero vectors, show that c bisects the angle between a and b, that is, divides it in half. Homework Equations none? The Attempt at a Solution dont know where to start manipulating?
  25. S

    Dot product within the commutator

    So I am reading a book on ferromagnetism, the author writes \left[S_J,S_{J-1} \cdot S_J\right] = S_J cross S_{J-1} (I couldn't get the cross product x in latex code for some reason) Where [tex]S_J[\tex] and [tex]S_{J-1}[\tex] are the spin operators for atoms j and j-1. I was...
  26. C

    Cylindrical and Cartesian Coord. dot product

    Homework Statement I'm given 2 unit vectors a_x and a_theta. I need to find the dot product between the two. Homework Equations Conversion from Cylindrical to Cartesian x = r * sin(theta) y = r * cos(theta) z = z Conversion from Cartesian to Cylindrical r = sqrt(x^2 +...
  27. D

    Integral with dot product in it

    Homework Statement I need to evaluate this integral: \int (\hat{r}\cdot\vec{a})\hat{r}d\Omega where \hat{r} is the radial unit vector, \vec{a} is a constant vector and d\Omega is solid angle element (\sin\theta d\theta d\phi). I saw something similar, but it was with tensors, and...
  28. P

    Dot product / vector magnitudes

    Homework Statement ||V|| = 2 ||W|| = 3 angle between vectors = 120 degrees V(dot)W = -3 Find ||2V+W|| Homework Equations Cos120 = V(dot)W / ||V|| ||W|| V(dot)V = ||V|| ^2 W(dot)W = ||W|| ^2 The Attempt at a Solution instead of solving for ||2V+W|| i instead try...
  29. B

    How Does the Dot Product Interact with the Gradient in Vector Calculus?

    dot product, and the gradient urgent pls!... Homework Statement Δ<-- this be the gradient and B<-- be a vector B X= xi +yj + zk *<---- be the dot product. (B*Δ)X=B Homework Equations n/a The Attempt at a Solution im not sure how to go about this but this is what i did i...
  30. P

    Taking the derivative of a dot product

    Homework Statement Hi, I need to take the derivative of the following energy potential (attached) in order to get an expression of the force so I can do a conjugate gradient minimization. An explanation of the potential is found in the attached jpeg below the functional form. Homework...
  31. L

    Find theta from the cross product and dot product of two vectors

    Homework Statement If the cross product of vector v cross vector w = 3i + j + 4k, and the dot product of vector v dot vector w = 4, and theta is the angle between vector v and vector w, find tan(theta) and theta. Homework Equations vector c = |v||w| sin(theta) where vector c is the...
  32. G

    What Does q*a Dot Product Mean?

    If someone says that you have to take the dot product of q*a and a is a length, and q is the 111 direction, or alonfg the 111 direction. What does that mean. Is it simply 111 dot length a. We have a exponential function to the power q dot a and want to know what to do.
  33. M

    Dot product - two points and a projection

    Homework Statement A person starts at coordinates (-2, 3) and arrived at coordinates (0, 6). If he began walking in the direction of the vector v=3i+2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. Homework...
  34. R

    Application of the Dot Product (Arfken and Weber)

    Homework Statement A pipe comes diagonally down the south wall of a building, making an angle for 45 degrees with the horizontal. Coming into a corner, the pipe turns and continues diagonally down a west facing wall, still making an angle of 45 degrees with the horizontal. What is the angle...
  35. B

    Integration by parts of a dot product scalar integrand

    Homework Statement Is this true or false? \int_V {\vec \nabla \Phi \bullet {\bf{E'}} \cdot {d^3}x} = \vec \nabla \Phi \bullet {\bf{E'}} - \int_V {\Phi \cdot \vec \nabla \bullet {\bf{E'}} \cdot {d^3}x}
  36. Q

    Dot product between cross products

    Homework Statement show (axb) . (cxd) = |a.c b.c| |a.d b.d| The Attempt at a Solution i have no idea. i don't know if the lines at the side are modulus lines or matrix brackets but it seems that it has something to do with distribution law. but i can't start proving if i...
  37. L

    Dot Product Question: Find Angle in Basketball Gym

    Homework Statement A basketball gymnasium is 25 meters high, 80 meters wide, and 200 meters long. For a half time stunt , the cheerleaders want to run two strings, one from each of the two corners above one basket to the diagonally opposite corners of the gym floor. What is the cosine of the...
  38. L

    Question about cross and dot product

    Homework Statement There are two points p1 pointing up p2 pointing rightHomework Equations The Attempt at a Solution I need to find \stackrel{\rightarrow}{p1} * r^ \stackrel{\rightarrow}{p2} X p1^ \stackrel{\rightarrow}{p2}* r^ they got \stackrel{->}{p1} * r^ =0 why is that? i know...
  39. N

    Dot product, cross product, Newton's laws.

    My final for Physics is coming up and I really, really ****ed up in this class. I don't even know how to do dot products, cross products, or let alone Newton's laws. We applied Newton's laws to some pulley problems I think. I've been going to all the lectures but it just doesn't stick -- nothing...
  40. R

    Dot product vector question

    Homework Statement i'm having troubles solving this problem in physics involving dot product rule Vector B is 5m and 60 degress above the x axis A has the same magnitude as C and C is has 25 more degrees than A .. find magnitude of A.. B dot C = 35 B dot A = 30 this all we're...
  41. C

    Proving perpendicular vectors using Dot Product

    Homework Statement Homework Equations dot product a.b = lal.lblcostheta vectors are perpendicular when a.b = 0 The Attempt at a Solution OA . LA = 0 OA = OM + MA LA = LO + OA OA . LA = (OM+MA) . (LO + OA) = OM.LO + OM.OA + MA.LO + MA.OAi get stuck here
  42. H

    Eigenvectors and the dot product

    Homework Statement Suppose the the matrix A is symmetric, meaning that A = a b b d Show that for any symmetric matrix A there are always real eigenvalues. Also, show that the eigenvectors corresponding to two di erent eigenvalues are always orthogonal; that is, if V1 and V2 are the...
  43. W

    Linear Algebra / Dot Product / matrices

    Homework Statement Let V=Rⁿ and let AεMnxn(R) Prove that <x,Ay> = <A^T x, y> for all x,yεV Homework Equations The Attempt at a Solution Can someone tell me if I'm on the right track? <x,Ay> = x^T Ay = (x^TA)y = <A^T x,Y>
  44. FeDeX_LaTeX

    Integration Involving Dot Product?

    Hello; How do I solve an integral involving the dot product? For example, imagine I have two vectors A and B, and I want to calculate; \int \vec{A}\cdot\vec{B} How do I do this? I am asking because, I read somewhere that I have to evaluate the dot product first: \int...
  45. S

    Visualizing the Dot Product Inequality of a, b & c in R^d

    Suppose I have three vectors a,b and c in R^d , And, I have that a.b < c.b(assume Euclidean inner product). What are the ways to visualize relation between a,b and c geometrically? I realize this is slightly open-ended, but am looking for insight here. Thanks in advance. PS: I have a thought...
  46. W

    Weird Dot Product Homework Q: Unusual Answer?

    Homework Statement i have the vector m(x(t), y(t)) = r = (s^t cos(t), e^t sin(t)) and want to find the line integral of itHomework Equations 1. \int m \centerdot r' dt 2. \int |m| |r'| dtThe Attempt at a Solution the answer is sqrt(2)/2 (e^pi - 1). when i do the problem the first way, i do not...
  47. I

    Linear algebra involving dot product and orthongal matrices

    Homework Statement Given: x (dot) y = x^T * y (where x,y are vectors; dot is dot product; and x^T is x transpose) and R is an orthogonal nxn matrix, and x,y are elements of R^n Show ||Rx|| = ||x|| The Attempt at a Solution I'm not sure what information I am suppose to use to solve...
  48. M

    Vectors dot product and cross product help

    Homework Statement Vectors A and B (both with the lines over it) lie in an xy plane. Vector A has magnitude 8 and angle 130 degrees, Vector B has components Bx=-7.72 and By=-9.2. a)What is 5(vector A) dot vector B? b)What is 4(Vector A) cross 3(vector B) in unit vector notation and magnitude...
  49. D

    Proving the Dot Product and Norm Theorems

    Homework Statement The Attempt at a Solution I let D be the center x = DX & a = DA (x-a) * (x+a)=|x|^2-|a|^2 Dunno what to do with the right side of the equation
  50. S

    Simple dot product [inner product] (verification)

    Homework Statement Homework Equations know dot product The Attempt at a Solution PART A PART B not sure what's it asking for help would be great
Back
Top