Dot product Definition and 390 Threads

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. L

    Dot product and the law of cosines

    I can't seem to derive the law of cosines from the vector addition of C = A + B using the dot product. Does anybody have any insights?
  2. S

    What is the meaning of Dot Product

    I just reviewed Dot Product, but I don't know what it actually, exactly means. would you tell me about its physical meaning or something interesting quality of it? Thanks
  3. T

    Calculating Electric Potential: That Tricky Dot Product

    I have become exceedingly confused by the various sign changes involved in computing the electric potential produced by a charge distribution, and I am sure I am simply forgetting a negative sign somewhere and am going crazy. However, V = -int from infty to r of E . ds ds is the...
  4. K

    Is the Dot Product Distributive Property Just Too Simple?

    Need help with this problem. I need to show that one side equals the other. I this is distributive property but isn't that just too plain simple? I mean I am in pre-cal class and therefore this question can't be easy as it seems... ::biggrin: anyways, if anyone knows about vector plez help...
  5. N

    What is the unit vector perpendicular to (2,1,-3) and (-1, 7, 4)?

    Use two methods to determine a unit vector perpendicular to both (2,1,-3) and (-1, 7, 4) Using Cross Product: Let v be a perpendicular vector to the given vectors v = a*b = (2,1,-3)(-1,7,4) =((1)(4)-(7)(-3), (-3)(-1)-(4)(2), (2)(7)-(-1)(1)) =(4+21, 3-8, 14+1) =(25, -5, 15) = (5, -1...
  6. R

    Dot product of electric and magnetic field conserved in special relativity?

    An inertial reference frame 2 is moving along the x-axis with constant velocity v with respect to inertial reference frame 1. ......->-> ->-> How can i prove the E.H = E'.H' ?? (dot product) using the 4 dimensional (Ax,Ay,Az,phi) where E = -1/c dA/dt - gradiant(phi) and H = curl(A) Where E is...
  7. J

    What Are the Basics of Calculating Cross Products in Vector Algebra?

    I was wondering if anyone could give me information the base of cross product all the problems I have been reading about so far have been in real depth I need to understand the basics first the problems given here are the beginning, if anyone can give me any help please do so, and thank you in...
  8. D

    Dot product cross product, where did they come from?

    While taking linear algebra I never really understood where the dot and cross product came from. It seemed that they were just randomly defined. I know that the dot product is an inner product but why is it defined as: \vec v \cdot \vec u = v_1 u_2 + v_2 u_2 ? why not \vec v \cdot \vec u =...
  9. A

    Finding Possible Values for K in Vector Algebra

    Hi I'm new here and I'm having trouble with this algebra question please help. Sorry if my latex is ugly I'm new to it. I get stuck at the bottom line and I'm not sure how to go further with the question to solve for K #7 - Angle between 2 vectors is \alpha where cos\alpha = \frac{3}{7}. a =...
  10. P

    How to Solve Cross and Dot Product Problems with u, v, and w?

    Ok, this one has really got me... Suppose that u \centerdot (vXw) = 2. Find (a) (uXv) \centerdot w (b) u\centerdot(wXv) (c) v\centerdot(uXw) (d) (uXv)\centerdotv Once I understand the what to do with the information given, I am sure the rest of the problems will fall into place...
  11. L

    Applications of vector dot product in physics

    I have to present shortly five applications of vector dot product in physics, for example: W=F*s. I am not quite clear about vectors, so could someone advise me.
  12. T

    Vectors with appl of dot product

    I'm trying to solve this problem with different approaches. Link is here http://img138.imageshack.us/img138/2268/problem1eb1.th.png the answer is 0i + 0.0927j + 0.0232 k lb* Means DOT Method one: Find the force in scalar components. Then Take the cross product of two position vectors to get...
  13. T

    Work done on a force using dot product

    The question with all the info is: A force F = (6x i + 5y j) N acts on an object as it moves in the x direction from the origin to x = 5.04 m. Find the work (work = integral of F (dot product) dx done on the object by the force. I am confused as to what dx would be in this equation, and how...
  14. J

    Vector question: dot product in cylindrical coordinates

    Hello. To find the work of a force, I have to perform a dot product between the force and a infinitesimal displacement. If they are in cylindrical coordinates, I can't manage to make the dot product. Please, could you help me? Thank you.
  15. F

    Dot product of basis vectors in orthogonal coordinate systems

    I'm doing a series of questions right now that is basically dealing with the dot and cross products of the basis vectors for cartesian, cylindrical, and spherical coordinate systems. I am stuck on \hat R \cdot \hat r right now. I'll try to explain my work, and the problem I am running into...
  16. H

    Prove Rectangle: Dot Product Help

    -Use the dot product to prove that the quadrilateral with vertices A(-4,1) B(4,5) C(7,-1) and D(-1,-5) is a rectangle. I tried find AB dot BC = 0, but I'm not getting 0. So I'm stumped. Can someone help me out? =/
  17. G

    Understanding the Derivative of a Dot Product: A Question on the Product Rule

    I have come across something that seems a little strange to me. The derivative of a dot product is something similar to the product rule. I am having difficulties grasping this. Isn't the dot product of two vectors a scalar? And then I always thought of a scalar as a real number and the...
  18. D

    Are Inner Products and Dot Products Fundamentally the Same?

    Is there any difference between an inner product and a dot product?
  19. infraray

    Dot Product Calculator Quandry

    I'm going nuts here and I can't figure this out. I have two complex vectors: A=(1+i)x + (1)y +(i)z B=(4-i)x +(0)y + (2-2i)z If I do the dot product of these on my calculator I get 1 + 7i, however when I do this by hand I keep getting 7 +5i. What am I doing wrong? When figuring this out by...
  20. E

    Rules regarding the vector cross product and dot product

    hi, I'm currently doing a mechanics module at Uni. The thing is, I'm not very sure about rules regarding the vector cross product and dot product. For example, it says in my notes for angular momentum: "Introducing polar coordinates \mathbf{r} = r(cos \Phi \mathbf{i} + sin \Phi...
  21. L

    How to: Force, work & dot product

    A force in the xy plane is given by: F= - (b/r3) (x i^ + y j^) where b is a positive constant and r= sqrt(x2 + y2) a) Show that the maginitude of the force varies as the inverse of the square of the distance to the origin and that its direction is antiparallel (opposite) to the radius vector...
  22. B

    Differentiation and dot product

    Can someone help me with the following question. I've been having trouble with problems of this kind for a while now. Q. If the path u(t) (u is a vector) is differentiable at least three times, simplify: \frac{d}{{dt}}\left[ {\left( {u' \times u''} \right) \bullet \left( {u' - u}...
  23. M

    Dot product with 3 dimensions, confused on concept, easy question i think

    Hello everyone! I'm confused on what I'm suppose to do here, I think i might got it though but i need to make sure... Here is the problem and my work: http://show.imagehosting.us/show/764032/0/nouser_764/T0_-1_764032.jpg he let r(t) = f(t) i + g(t) j + h(t) k. So if i multiply this by...
  24. M

    Dot product, i don't see what they want from me

    hello everyone, i understand the dot product and its properties but i don't get what they want! They say... The dot product of two vectors are perpendicular if a.b = 0. Then any vector in R^3 perpendicular to -5 -6 -3 note: that is a matrix above. can be written in the form...
  25. D

    Geometric Proof of Dot Product: |A dot B| ≤ |A||B|

    I am doing a assingment for my classical mechanics class that requires the proof of: The dot product of |A dot B| <= (less than or equeal to) |A| |B| . I did the algebraic proof fine but we are required to do a geometic proof as well. This leaves me with the question what is the geometic...
  26. M

    Showing dot product is commutative, please

    Hello everyone. I'm trying to show ab = ba. //communative property of dot product This is what I have, is it enough to show this? ab = (ax, ay, az) \dot (bx, by, bz) = axbx + ayby + azbz; a\dotb = (axi + ayj) (bxi + byj); //note: i and j are unit vectors = axbx(i)(i) + ayby(j)(j) +...
  27. M

    Conceptual explanation of Dot Product

    I'm a peer leader for a general physics lab and someone asked me to explain what the Dot Product meant conceptually. I told him it was the projection of A onto B multiplied by the magnitude of B. He looked even more confused after that; my questions are: a) Did I explain it correctly...
  28. M

    Show the commutative property with dot product

    Hello everyone, does anyone know the proof of the dot products communative property (a)(b) = (b)(a) or any websites that show the dot products communative property? or other properties? Thanks! The book only shows the distributed property.
  29. D

    Understanding Dot Product: Identity and Unit Vectors

    What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity? Thanks, don_anon25
  30. M

    Integration involving a dot product

    I am trying to do the following vector integral. When the force and the initial motion are along the same line it is easy but when they are not I can't do the integration. Hints would be appreciated. \vec F = \frac {d} {dt} \vec p where \vec F = (0, F_{y},0) and \vec {p} can be...
  31. S

    Learning vectors: the dot product of vectors

    Well, I've been attempting to learn dot products of vectors and in doing so have come upon some questions. NOTE: a, b, and c are to be regarded as vectors Please regard '*' as the dot for multiplication, too. Question 1: a * b * c When I see this, I am unaware how to approach it...
  32. J

    Simplifying Dot Product Equations with Vectors - Need Help!

    i need help with the following: note that the big dots represents the dot product 1. suppose that a \bullet b = c \bullet b for all vectors \overrightarrow{b} . show that \overrightarrow{a} = \overrightarrow{c} . i suppose i can't simply divide out the b, right? anyway, i tried...
  33. M

    What Are Some Dot Product Questions in Geometry and Discrete Class?

    I have some questions from my geometry and discrete class.. #1 Given a and b unit vectors, a) if the angle between them in 60 degrees, calculate (6a+b)●(a-2b) b) if |a+b|= root(3), determine (2a-5b)●(b+3a) #2 The vectors a=3i-4i-k and b=2i +3j-6k are the daignols of a parallelogram...
  34. RadiationX

    Understanding Vector Dot Product: Solving for a, b, and c in a Right Triangle

    i have three vectors: a=4,b=3,c=5 that form a right triangle. vector a is in the positive x direction, vector b is in the positive y direction starting at the tip of a. vector c is the hypotenuse of the triangle with tip at the origin. (see attaced picture .doc file) the questions are...
  35. M

    What Is the Unit Vector Perpendicular to Both 4i - 3j + k and the z-Axis?

    I'm sorry if this is placed under the wrong section of the forum. But i really need help with a problem. Well, here it is; Find a unit vector that is parallel to the xy-plane and perpendicular to the vector 4i - 3j + k *note* there is a ^ above the ijk.
  36. B

    Exploring Perpendicularity in 2D and 3D

    Let n = (a,b,c) v = (x,y,z) Whatever the dot product of these vectors equal to, let's call d, the vector n is perpendicular to v. Again, I cannot stay calm and ask WHY? If we call n = (a,b); v= (x,y) ==> ax+by= d and the slope of this line is -a/b whereas the slope of the vector n is...
  37. C

    Calculating the Dot Product: A*B

    I need some major help with some dot products. I was curious if when getting the dot product, you add the angles of each one to each other. I'll give the problem: theta=(/) Find A*B(dot product) A=8.6i+5j B=9.7i+2.6j So i used the formula my teacher gave me to find the angle, theta...
  38. D

    Understanding the Units of the Dot Product

    Hi, I got a simple question, "dot product" have units? I mean, if A=(Ax+Ay+Az)N and B=(Bx+By+Bz)(cm/s) , the units of A.B will be N.(cm/s) Thanks, Cali
  39. P

    Find p Given a.b = 4, a=[6,3,-2], b=[-2,p,-4]

    The angle between vectors a and b is cos^-1(4/21). Find p if a = [6,3,-2] and b = [-2, p, -4] I did: cos x= 4/21 = a.b/|a||b| The result comes out to be p=8/3 but it only satisfies that a dot b is 4 and not |a||b| = 21...What am I doing wrong?
  40. Juntao

    Help with simple dot product proof

    Here's what I got to prove where '.' is dot. A.B=A.C Then B=C True or false? If true, prove it in general terms, if false, provide a counter-example. Ok, I just need some body to comment on my little proof here, and any guidelines to make it more thorough or whatnot. I know that the dot...
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