# Dot product Definition and 38 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. ### I Dot product of unit vectors

HI, I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ Why is it like that? Thanks
2. ### Dot product vectors problem

Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...
3. ### I Why is (N dot N) different for magnitude than for X, Y, Z components?

I have to perform a calculation on my data. Here is an example of data from just one time step (data from other time steps would appear as additional rows). X Y Z Total 2 2 1 3 Total = SQRT(X2 + Y2 + Z2). The calculation I have to do is: (N • N), where "N" is an average. I tried...
4. ### I Proof concerning the Four Fundamental Spaces

Hello all, I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the row space to the null space column space to the left null space ------------------------------------------------ In the book of G. Strang there is this nice picture...
5. ### I Dot product in spherical coordinates

I'm learing about antennas in a course, and we are using Jin's Electromagnetic text. This isn't a homework problem, I'm just trying to understand what I'm supposed to do in this situation. This part of the text discusses how to evaluate a radiation pattern. One of the steps to evaluate the...
6. ### Proof of a dot product using sigma notation

Mentor note: Moved from a technical section, so is missing the homework template. Hi, I'm always not sure how to prove something in math and I'm wondering if this is enough. ##\vec r \cdot (\vec u + \vec v) ## ##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s## ##\vec r \cdot (\vec u +...
7. ### I Why should a Fourier transform not be a change of basis?

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...
8. ### I Dot product in Euclidean Space

Hello As you know, the geometric definition of the dot product of two vectors is the product of their norms, and the cosine of the angle between them. (The algebraic one makes it the sum of the product of the components in Cartesian coordinates.) I have often read that this holds for Euclidean...
9. ### Determining the power of frictional force

I can say that the frictional force always against the rolling sphere and the velocity is increasing for the ball. So The dot product F.v keeps on getting more and more negative, so how can the Pf remain constant? Well the velocity increases along the incline and the force of gravity is down...
10. ### I Dot Product with Derivative

Summary: The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector 'b'. Hello, I have the following Problem. The Dot Product of a Vector 'a' and a Derivative 'b' is the same like the negative of the Derivative 'a' and the Vector...
11. ### Stuck on a few Vector homework problems

I'm stuck on a few Vector homework problems. I don't quite understand how to write vectors A+B and A-B for questions 1b and 2b. I tried starting with calculating the magnitude for vector A+B on question 1b and then followed by finding theta, but I'm not sure if that's what I'm supposed to do...
12. ### Python Invert a matrix from a 4D array : equivalence or difference with indexes

I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
13. ### B Line Integral, Dot Product Confusion

From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
14. ### B Dot product scalar distributivity

I'm having a little trouble with this : We have ##(\alpha\vec{a})\cdot b = \alpha(\vec{a}\cdot\vec{b})## but shouldn't it be ##|\alpha|(\vec{a}\cdot\vec{b})## instead since ##||\alpha \vec{a}||=|\alpha|.||\vec{a}||## ? ##(\alpha\vec{a})\cdot b = ||\alpha\vec{a}||.||\vec{b}||.\cos\theta =...

23. ### Angle between vector and tangent vector

Homework Statement My problem is: For the logarithmic spiral R(t) = (e^t cost, e^t sint), show that the angle between R(t) and the tangent vector at R(t) is independent of t. Homework Equations N/A The Attempt at a Solution The tangent vector is just the vector that you get when you take the...
24. ### I Validity of proof of Cauchy-Schwarz inequality

Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero. If neither x nor y is zero, then by x · y = | x | | y | cos θ, |x · y|=| x | | y | cos θ | ≤ | x | | y | since -1 ≤ cos θ ≤ 1 How valid is this a proof of the...
25. ### Question about determining the angles of triangle given two vectors

<<Mentor note: Missing template due to originally being posted elsewhere>> Hello everyone. I have the following problem: Determine the angles of a triangle where two sides of a triangle are formed by the vectors A = 3i -4j -k and B=4i -j + 3k I thought that I would find the third side being...
26. ### I The "real" angle between two triangular surfaces

Hello everyone, i'm new to the forum so hope it is the right place for my question :) i need to know the angle between two triangular surfaces, the easiest way would be extract the normal for each surface(u,v) and then using the dot product we can easily compute the cosine for the angle I'm...
27. ### Can someone help me find the angle between two forces?

Can someone help me relearn finding the angle between two forces when solving for work of each forces (gravity, tension, fF, normal)? I remember that cos(90°-α) = sin(α) but what I don't understand is when the angle in between is "90°-α" or when it's just "α". I tried doing this on my own and...
28. ### What is the largest number of mutually obtuse vectors in Rn?

This is my question: What is the largest m such that there exist v1, ... ,vm ∈ ℝn such that for all i and j, if 1 ≤ i < j ≤ m, then ≤ vi⋅vj = 0 I found a couple of solutions online. http://mathoverflow.net/questions/31436/largest-number-of-vectors-with-pairwise-negative-dot-product...
29. ### Dot Product

Homework Statement Homework Equations p.q+p.r The Attempt at a Solution I've expanded p.(q+r) to give p.q+p.r. The magnitude of p is 3, and since ABE is an equilateral triangle, the magnitude of q is also 3, right? So then p.q=9, but the answer scheme states that p.q=4.5. I'm still pretty...
30. ### Cauchy's equation in terms of material acceleration

Does anyone know which formula is used or how to arrive at the righthand side of the equation below, which is the dot product of del and rho*a 2nd order tensor(V V). . represents dot product and X a vector quantity This problem is in connection with transforming cauchy's equation in terms of...
31. ### Why work done by a force is a scalar product

Why work done by a force was taken as dot product between force applied and displacement caused?
32. ### Proving volume of box using cross and dot product

Homework Statement The diagram shows a box with parallel faces. Two of the faces are trapezoids and four of the faces are rectangles. The vectors A, B, and C lie along the edges as shown, and their magnitudes are the lengths of the edges. Define the necessary additional symbols and prove...
33. ### Index Notation help

1. The problem is: ( a x b )⋅[( b x c ) x ( c x a )] = [a,b,c]^2 = [ a⋅( b x c )]^2 I am supposed to solve this using index notation... and I am having some problems. 2. Homework Equations : I guess I just don't understand the finer points of index notation. Every time I think I am getting...
34. ### Dot product vs trigonometry in Gauss' law

I'm currently writing my EP on various physical equations including Maxwell's equations, and I had to justify using the dot product of the normal unit vector and the electric field in the integral version. However, I can't think of a reason for not using trigonometry as opposed to the...
35. ### Multiple Integral Challenge Question, I just need a hint

Homework Statement I will just post an image of the problem and here's the link if the above is too small: http://i.imgur.com/JB6FEog.png?1 Homework Equations The Attempt at a Solution I've been playing with it, but I can't figure out a good way to "grip" this problem. I can see some...
36. ### Given two vectors, find vector of the parallelogram height

Homework Statement Find the coordinates of the vector of the height of the parallelogram formed by vectors a={1, 2, 1} and b={2, -1, 0} Homework Equations A=|axb|, A=|a|*h The Attempt at a Solution I can find the intensity of the vector h i.e the length of the height, but not its vector. I...
37. ### 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...
38. ### 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...