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. O

    Can the Dot Product of Two Unit Vectors Ever Be Less Than 1?

    I need certain stopping criterion for approximating one unit vector with another. In case there is a perfect match (after a number of iterations), the dot product of the vectors is 1. I need to know (and have a reasoning for) whether in any other case the dot product of the original unit vector...
  2. B

    Dot Product Question, Perpendicular

    pretty much this is all i was given. I have no idea how to even approach it. I do not need an answer (would be nice though), just an idea on how to go about starting it.
  3. L

    Dot Product Rules Multiple choice

    Homework Statement See the image http://tinypic.com/r/33z6vz5/7 Homework Equations The rules of dot product multiplication. The Attempt at a Solution I've determined that 2 of the solutions are valid, while two of the choices are invalid. I thought C and D looked correct, making A...
  4. T

    Conditions of dot product

    Homework Statement Under what conditions are the following true? a) |\mathbf{a}\cdot \mathbf{b}| =|\mathbf{a}||\mathbf{b}| b) (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = |\mathbf{a}|^2 - |\mathbf{b}|^2 Homework Equations None. The Attempt at a Solution a) I...
  5. S

    Linear algebra with dot product?

    Homework Statement Let A(theta)x = b for each theta in S. Calculate, (x dot b)/(|x||b|) A = [cos(theta) -sin(theta) sin(theta) cos(theta)] How is this related to theta? Recall that x dot y and |x| are the standard dot-product and magnitude, respectively, from vector-calculus...
  6. N

    Dot product wagon incline

    A person pulls a wagon up an incline that makes an angled of 30 deg with the horizontal with a force on the handle of 30 pounds which makes an angel of 30 deg with the incline find the work done in pulling the wagon 100 ft 20cos30 will be a force along the surface of the incline and since...
  7. Z

    Vector dot product homework help?

    if A = 2i^+6k^ B=6i^+5j^ find A.B if A.B=0 then what do u conclude from it? one more question if A = 6i^+9k^ and B=-4i^+4j then find A+B,A-B,A.B
  8. S

    Integral of Complex exp of dot product

    S(\vec{q})= \int_0^r exp(i\vec{q}\cdot\vec{x})4\pi x^2 \ dx How would one approach this integral? I tried to "ignore" the dot product and proceeded with exp(i\vec{q}\cdot \vec{x})=exp(iqx) and got a wrong answer.
  9. K

    Prove Vector Derivative (using Dot Product)

    Homework Statement n. Let r1 and r2 be differentiable 3-space vector-valued functions. Directly from the definition of dot product, and the definition of derivative of vector- valued functions in terms of components, prove that d/dt (r1(t) • r2(t)) = r′1(t) • r2(t) + r1(t) • r′2(t). Homework...
  10. N

    Proof of Dot Product: X*X & X=0 Questions

    If X is an N vector, is it 1) possible for X*X to be negative? 2) if x*x=0, what is X. I am having trouble writing the proper proof. for 1 I stated that it is impossible for X*X to be negative bc if x is positive, X*X is positive and if x is negative, -X*-X is still positive. for 2 I...
  11. Q

    Vector product VS dot product in matrix

    hi, i don't really understand what's the difference between vector product and dot product in matrix form. for example (1 2) X (1 2) (3 4) (3 4) = ? so when i take rows multiply by columns, to get a 2x2 matrix, i am doing vector product? so what then is dot producT? lastly...
  12. 1

    Mastering Cross and Dot Products: A Guide for Test Review

    i'm reviewing for a test and I can't remember how to do the cross product of (2A)x(3B) or how to find the angle theta when given components. or dot product: 2A . 3B
  13. S

    Is Triangle ABC a Right Triangle?

    Homework Statement Given triangle ABC with vertices A(4, 1, 7), B(-2, 1, 1) and C(-3, 5, -6)..is ABC a Right triangle Homework Equations The Attempt at a Solution I took the dot product of vertices A(4, 1, 7).B(-2, 1, 1), and it gives 0..but however I am a little confused, as...
  14. S

    In this case, the correct dot product is 22a.a + 15a.b - 3b.b.

    Homework Statement Expand and simplify: (2\vec{a} + 3\vec{b}) .(5\vec{a} - \vec{b} Homework Equations The Attempt at a Solution I tried expanding it, but was a little confused, and would really appreciate any help..would it be (2\vec{a})(\vec{a}) = 2 \vec{a}2? Thanks..
  15. 8

    Dot Product of G and A: Where Do the Denominators Go?

    Homework Statement what is the dot product of G{\bullet}A where A = \left(\frac{a_3}{l}-\frac{{a_1}}{h}\right) and G = 2{\pi}h{\frac{{a_2}}x{{a_3}}}{{a_1}{\bullet}{{a_2}}x{{a_3}}} Homework Equations The Attempt at a Solution The answer is zero and I've got the worked solution...
  16. W

    Dot product and cross product of vectors

    Homework Statement This isn't so much a problem of calculation, so much as it is I need to know if i did it right, you'll see what I mean. vectors v=(2,0,2), u=(-1,1,0), w=(0,-1,1) Computer the quantities that make sense (a period denotes the dot product, x denotes cross product)...
  17. V

    What are the values of s that make two given vectors orthogonal?

    By evaluating the dot product, find the values of the scalar s for which the two vectors b=X+sY and c=X-sY are orthogonal also explain your answers with a sketch: my working (X,sY).(X,-sY) has to equal 0 for them to be orthogonal x.x = 1 since they are unit vectors...
  18. E

    Exploring the Dot Product Expansion: Proving the Relationship between a and b

    Homework Statement Show that the dot product of vectors a and b is equal to 1/4|a+b|^2 - 1/4|a-b|^2 Homework Equations a dot b = |a||b|cos(theta) a dot b = a1b1 + a2b2 + ... The Attempt at a Solution I've tried using the combination of the cosine law and those two above dot...
  19. H

    Determinant, dot product, and cross product

    I am reading through David Widder's Advanced Calculus and he abbreviates a determinant as: \left( \begin{array}{cccc} r_{1} \ s_{1} \ t_{1}\\ r_{2} \ s_{2} \ t_{2}\\ r_{3} \ s_{3} \ t_{3}\\ \end{array} \right) And refers to it by (rst). He then states...
  20. C

    Dot product of vector and symmetric linear map?

    Homework Statement My book states as follows: --- If u and v have the coordinate vectors X and Y respectively in a given orthonormal basis, and the symmetric, linear map \Gamma has the matrix A in the same basis, then \Gamma(u) and \Gamma(v) have the coordinates AX and AY, respectively. This...
  21. L

    Dot Product: Explaining 1/2 Coefficient

    OK, this has been bugging me for a while. Why is it that \mathbf{v} \cdot \frac{d \mathbf{v}}{d t} = 1/2 v^2 where regular v is just the magnitude of bold v or more specifically where does the 1/2 coefficient turn up.
  22. Rasalhague

    Dot product of 4-force and 4-velocity

    In what follows, I'll use bold for 4-vectors and an arrow over a letter to denote a 3-vector. I'll use a dot both for the Minkowski dot product and the Euclidean one, and multiplication of real numbers; the meaning in each case should be clear from the symbols on either side. \vec{v} and \vec{a}...
  23. T

    Dot product as matrix products?

    Why not make dot(u,v)=transpose(u)v rather than transpose(v)u?
  24. M

    Cross product and dot product of forces expressed as complex numbers

    Homework Statement I have came up with an example to illustrate my question. There is a rod, which can turn around p1. p1p2 = (-1+j) m p1p3 = (-3 + 3j) m p1p4 = (1 - j ) m F1 = (1+3j) N F3 = (-1 - 2j ) N F4 = unknown, orthogonal to the rod compute F2_n, orthogonal component of F2 to the...
  25. E

    Matrix multiplication and the dot product

    Homework Statement Suppose that A is an n x n matrix such that A(Transpose)A=I. Let x be any vector in R^n. Show that llAxll=llxll; that is, multiplication of x by A produces a vector Ax having the same length as x. Homework Equations Sqrt(x(transpose)x)=llxll The Attempt at a...
  26. T

    Dot product and cross product evaluation questions

    This question has a few parts. r = i + 2j + 3k s = 2i - 2j - 5k t = i - 3j - k Evaluate: a)(r.t)s - (s.t)r b)(r x s) x t. deduce that (r.t)s - (s.t)r = (r x s) x t can you prove this relative true for any three vectors a)(r.t)s - (s.t)r (r.t)s well I don't know...
  27. T

    Question about vectors and dot product.

    Hi, Can someone tell me if: -E dotted with ( A + B ) is equal to -E.A -E.B where E, A and B are all vectors What I mean is, does the minus sign appear on the E.B bit as well? Also, is \int \frac{d}{dt} (A) dV equal to: \frac{d}{dt} \int (A) dV Thank you
  28. J

    Understanding Products in the Equation v^2 = u^2 + 2aS

    In the equation v^2 = u^2 + 2aS , What kind of products are v^2 , u^2 , and aS ? Cross product or dot product?
  29. O

    Dot Product, Derivative, and Vector Valued Functions

    Homework Statement Prove that, if \vec{r}(t) is a differentiable vector valued function, then so is ||\vec{r}(t)||, and \vec{r}(t) \bullet \vec{r'}(t) = ||\vec{r}(t)|| ||\vec{r}(t)||'Homework Equations I know how to do a dot product, but what bothers me is the fact that the question involves...
  30. W

    Need help with combination of dot product and cross product question

    Homework Statement Let \vec{A}, \vec{B}, and \vec{C} be three vectors which are all not in the same plane. Show that \vec{A}{\cdot}(\vec{B}{\times}\vec{C})=\vec{B}{\cdot}(\vec{C}{\times}\vec{A})=\vec{C}{\cdot}(\vec{A}{\times}\vec{B}) Homework Equations Don't know :( The Attempt at...
  31. A

    What is the Meaning of the Dot Product in Integrals?

    What does the following mean - \int \textbf{F} \cdot d \textbf{r} I know that its equal to work done. but i have problem in understanding it, what's the varibale. i am familiar with integrate cosxdx. what's is similar to dx in above equation. is it ds. Now how can one get to this from above...
  32. M

    Confusion regarding dot product of vectors(row matrices)

    I’ve got a confusion. We know a 1x3 row matrix is a 3-vector i.e. x= [ a b c] Matrix x can be written in vector notation like x= a i + b j + c k where i, j, k are unit vectors along x,y & z axes. For dot product of x.x = a2 + b2 + c2 when x= a i + b j + c k But according to...
  33. T

    Dot product and cross product problem

    I am having problems with this questions my physics teacher gave us. Since he doesn't explain things well, I do not know how he got an answer of 66. Can you please explain to me how to get this answer? THX. For the following three vectors, what is C*(A X 3B) (the "*" is suppose to be a dot)...
  34. T

    Understanding the Dot Product of Derivatives

    Hi, Im having trouble understanding something in one of my Dynamics lectures. The lecturer said that: dr/dt dotted with d2r/dt2 (where r is a vector) equals: (1/2)(d/dt(dr/dt dotted with dr/dt))... I just can't get this result. I know it has something to do with the product rule...
  35. W

    Help with dot product for vectors

    Question Let vectors \vec{A} =(2,1,-4), \vec{B}=(-3,0,1), \vec{C}=(-1,-1,2). What is the angle (in radians) \theta_{AB} between \vec{A} and \vec{B}?Important equations \vec{A} \cdot \vec{B} = \vert \vec{A} \vert \,\vert \vec{B}\vert \cos(\theta), where \theta is the angle between \vec{A} and...
  36. X

    What is the Geometric Interpretation of Dot and Cross Products?

    Describe the surface defined by the equation: (a) \vec{r}\cdot \hat{x}= 2, where \vec{r}=x\hat{x}+y\hat{y}+z\hat{z}; (b) \left \| \vec{r} \times \hat{x}\right \|=2 For the first one, I know that is interpreted as the projection of the r vector onto the x-axis is equal to two...
  37. L

    Dot product of vector function?

    Greetings. I was thinking about finding the angle between two functions, so I thought it may be elegant to turn them into vector valued functions, and find the dot product at a given variable value where the vectors lie on the same plane and are functions of the same variable. I'm going to go...
  38. G

    Dot Product of Momentum and Radial Operators

    Homework Statement I need to find the momentum space function for the ground state of hydrogen (l=m=0, Z=n=1) Homework Equations \phi(\vec{p}) = \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i(\vec{p}\cdot\vec{r})/\hbar}\psi(\vec{r})d^3\vec{r}...
  39. C

    Can all dot product computations be computed?

    Homework Statement Which of the following can be computed? 1. A dot B dot C 2. A dot ( B dot C ) 3. A dot ( B + C ) 4. 3 dot A Homework Equations The Attempt at a Solution I believe that 2 and 3 are the only two that can be computed. Can anyone confirm this? Thanks.
  40. C

    Dot Product Notation Clarification

    The following notation is from the book "Frames and Bases." Let f and g be vectors in R^{n} with the usual dot product <,>. Then, what does the notation \left|\left\langle f,g\right\rangle\right|^{2} mean? Specifically, does it mean \left|\sum^{n}_{i=1}f_{i} g_{i}\right| or does it...
  41. T

    Cross product, dot product concepts

    Hi, I'm very ashamed about getting a fully understanding of these vector product concepts. But i did a lot of search and get an idea about them. I read Feynman's lecture on physics and almost completely understand the mathematics behind their properties. In that book feynman proves that cross...
  42. Д

    Geometrical interpretation of dot product (trying to prove)

    Hi! I am trying to find out where: cos\theta=\frac{a \cdot b}{|a||b|} came from. Here is mine geometrical interpretation of scalar projection: Now, (pr means projection) pr_{\overrightarrow{A}} \overrightarrow{B} = p\overrightarrow{B_0} and...
  43. F

    Determining Vector Dot Product Using Unit Vectors

    Hi All, I'm currently studying vector projections and the vector dot product. I ran into a problem on the homework I wasn't quite sure how to tackle... Any suggestions?The problem is stated as follows: Determine ||v+w|| if v and w are unit vectors separated by an angle of \frac{\pi}{6}...
  44. K

    Finding Angle in 3D using dot product

    Homework Statement Shown are a mast and a portion of the rigging of a schooner. Members CD and EF lie in the same plane, and CD is of length 7.5 m and forms an angle of 45° with a vertical line drawn through C. Knowing that when \theta = 45° the tension in rope BD is 250N, determine, (a) the...
  45. P

    Dot Product Issues Homework: Vector A and 4 Vectors

    Homework Statement Vector A and four other vectors that have the same magnitude but differ in orientation. a) Which of those other four vectors have the same dot product with A? b) Which have a negative dot product with A? http://img195.imageshack.us/img195/7079/40191924.th.jpg (Those...
  46. A

    Faraday's Law, Magnetic Flux, and the dot product

    Homework Statement We are studying Electromagnetic Induction right now. I understand the concepts, Faraday's Law and magnetic flux. But I don't understand what my book is doing. Homework Equations Magnetic Flux \phi=\intB∙dA Faraday's Law Emf = - d\phi/dt Emf=Electromotive force \phi=Magnetic...
  47. I

    Solving for p in a Vector Dot Product Equation

    Homework Statement The cosine of the angle between vector a and vector b is 4/21. Find p. vector a = 6i + 3j - 2k vector b = -2i + pj - 4k Homework Equations (Vector a) . (Vector b) = abcos(theta) (Vector a) . (Vector b) = axbx + ayby + azbz mag(a) = root(ax^2 + ay^2 + az^2) The Attempt at...
  48. R

    Calculating Orthogonality of Binormal Vector with Dot Product

    How can I show that the binormal vector is orthogonal to the tangent and normal vector. I know i should use the dot product to determine this, however i do i actually go about doing it?
  49. L

    Maths questions on dot product, vectors?

    Homework Statement 1. (a) Define carefully the dot and vector products of two vectors a and b. (b) Show, using the dot product, that if c - d and c + d are perpendicular then |c| = |d|. (c) The vectors a = i+2j and b = i - 2j + k form two sides of a triangle. Use vector methods to find...
  50. R

    The Dot Product and Cross Product: Finding the Angle Between Two Vectors

    Homework Statement If theta is the angle between two non-zero vectors A and B, then which of the following angles theta results in A dot B = |A x B|? Homework Equations A dot B = ABcos(theta) A x B = ABsin(theta) The Attempt at a Solution There were two choices in the multiple choice answers...
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