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.
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
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)
-...
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...
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...
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...
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 +...
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...
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...
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...
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...
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...
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) ...
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...
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 =...
This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam:
There are two cases but the excercise is pretty much the same:
Compute
$$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
Homework Statement
Given that vector a = (1, 2, -5), b = (-12, 41, 75) and c = a + 2b, explain why (without doing any calculations whatsoever) the value of a•(b x c) = 0
Homework Equations
No specific equations, as the question asks for the value without making any calculations. This problem...
I'm trying to work through a scattering calculation in the Peskin QFT textbook in chapter 5, specifically getting equation 5.10. They take two bracketed terms
4[p'^{\mu}p^{\nu}+p'^{\nu}p^{\mu}-g^{\mu\nu}(p \cdot p'+m_e^2)]
and
4[k_{\mu}k'_{\nu}+k_{\nu}k'_{\mu}-g_{\mu\nu}(k \cdot...
Start with a circle of radius r and center c. Inside of that circle is an arbitrary point p. Given an arbitrary normalized direction vector d, I need to find the radius and center the circle that (1) intersects p, (2) is tangent with the circle centered at c, and (3) has its center lying on the...
Hello all!
I usually don't like to ask for help... But this is the first week of courses and I'm already stumped on a homework question...
1. Homework Statement
So the question states: Find the work by the force F = x^i + xy^j. If the object starts from the origin (0,0), moves along the...
Chansu
Thread
dotproduct
integration
vector
vector components
work
Hi all,
Suppose we have vectors coming in order as A, B and then C (but A must be deleted before C comes in). Then how to get the dot product between A and C? It is allowed to store some calculations of A before deleting elements of A, for example, we could store norm of A, dot(A, B) and etc...
In 'Introduction to Electrodynamics' by Griffiths, in the section of explaining the Gradient operator, it is stated a theorem of partial derivatives is:
$$ dT = (\delta T / \delta x) \delta x + (\delta T / \delta y) \delta y + (\delta T / \delta z) \delta z $$
Further he goes onto say:
$$ dT =...
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...
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...
<<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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...