What is Scalar product: Definition and 91 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.
Hi,
from Penrose book "The Road to Reality" it seems to me inner product and dot/scalar product are actually different things.
Given a vector space ##V## an inner product ## \langle . | . \rangle## is defined between elements (i.e. vectors) of the vector space ##V## itself. Differently...
Hi everyone,
I'm having problems with task c
In the task, the norm has already been defined, i.e. ##||\vec{c}||=\sqrt{\langle \vec{c}, \vec{c} \rangle }## I therefore first wanted to calculate the scalar product of the cross product, i.e. ##\langle \vec{a} \times \vec{b} , \vec{a} \times...
Hi,
unfortunately, I have problems with the task c
I used the tip with the Laplace evolution theorem and rewrote the determinant to calculate ##W_n##. Then I simply formed the scalar product ##W_1 W_n## and here I get now no further.
Hi,
Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have...
Hi
If i have 2 general vectors written in Cartesian coordinates then the scalar product a.b can be written as aibi because the basis vectors are an orthonormal basis.
In Hamiltonian mechanics i have seen the Hamiltonian written as H = pivi - L where L is the lagrangian and v is the time...
I know that taking the scalar product of the harmonic (Laplacian) friction term with ##\underline u## is
$$\underline u \cdot [\nabla \cdot(A\nabla \underline u)] = \nabla \cdot (\underline u A \nabla \underline u) - A (\nabla \underline u )^2 $$
where ##\underline u = (u,v)## and ##A## is a...
Hi,
This feels like such a stupid question, but it's bugging me. Two displacements can be represented with two vectors. Let's say their magnitudes are expressed in metres. The scalar (dot) product of the two vectors results in a value with the units of square metres, which must be an area. Can...
Homework Statement
a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself.
b: What would be the r, θ components of the unit vector in the θ direction?
Homework Equations
Scalar product of 2 vectors = AαgαβBβ
The Attempt at a Solution
For part a, I used the...
I'm reading up on the Lagrangian equation, but what I'm asking is to do with electromagnetism.
In the first equation here: http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf
L equals the kinetic minus the potential energy. For the potential energy term, I just don't...
1) Firstly, in the context of a dot product with Einstein notation :
$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$
with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
In Schutz's A First Course in General Relativity (second edition, page 45, in the context of special relativity) he gives the scalar product of four basis vectors in a frame as follows:
$$\vec{e}_{0}\cdot\vec{e}_{0}=-1,$$...
Hi,
I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2)
is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...
Hi everyone !
I would like to know the real meaning of scalar product. So, I know scalar product is defined as :
||a||.||b||.cos(a;b) = k
But what k is ?
(Sorry for my english, I am french).
Regards :)
Alright, so we ran into a peculiarity in answering this question.
Let R be the set of all functions f defined on the interval [0,1] such that -
(1) f(t) is nonzero at no more than countably many points t1, t2, . . .
(2) Σi = 1 to ∞ f2(ti) < ∞ .
Define addition of elements and multiplication...
It is very well known result that ##grad[e^{i\vec{k}\cdot \vec{r}}]=i\vec{k}e^{i\vec{k}\cdot \vec{r}}##. Also ##\vec{k}\cdot \vec{r}=kr\cos \theta## and ##gradf(r)=\frac{df}{dr} grad r##. Then I can write
grad e^{ikr\cos \theta}=ik\cos \theta e^{i \vec{k}\cdot \vec{r}}...
Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product?
Thanks!
Andy
Hello, I have a question about why I can't determine the angle between two vectors using their cross product.
Say there are two vectors in the XY-plane that we want to find the angle between:
A = -2.00i + 6.00j
B = 2.00i - 3.00j
The method to do this would be to work out the scalar product of...
Let's say I have two vector fields a(x,y,z) and b(x,y,z).
Let's say I have a scalar field f equal to a•b.
I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b.
Ideally, I would like to be able to express...
Consider $X, Y$ as $n \times n$ matrices, I am given this definition of scalar product:
$$\langle X, Y \rangle = tr(X Y^T),$$
and I need to prove that it is positive definite scalar product. Of several properties I need to prove, two of them are
(1) $\langle X, X\rangle \geq 0$ and
(2)...
Hey guys,
I got the following derivation for some physical stuff (the derivation itself is just math)
http://thesis.library.caltech.edu/5215/12/12appendixD.pdf
I understand everything until D.8.
After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...
I recently learned that the general formula for the dot product between two vectors A and B is:
gμνAμBν
Well, I now have a few questions:
1. We know how in Cartesian coordinates, the dot product between a vector and itself (in other words A ⋅ A) is equal to the square of the magnitude |A|2...
Scalar Product is defined as ##\mathbf A \cdot \mathbf B = | \vec A | | \vec B | \cos \theta##.
With the construct of a triangle, the Law of Cosines is proved.
##\mathbf A## points to the tail of ##\mathbf B##.
Well, ##\mathbf C## starts from the tail of ##\mathbf A## and points to somewhere...
Why A.A = ||A||^2 , I know that from product rule we can prove this where theta =0 , I am asking this because I have seen many proves for A.B = ||A||||B||cos(theta) and to prove this they have used A.A = ||A||^2, how can they use this , this is the result of dot product formula. I havee seen...
why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove...
Homework Statement
Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
Homework Statement
Refer to solution II , the author used the scalar analysis( dot product) to get the direction of moment ...IMO , this is incorrect ... Only cross product can be determined this way . correct me if I'm wrong .
Homework EquationsThe Attempt at a Solution
From a textbook. proof that the scalar product ##A\centerdot B## is a scalar:
Vectors A' and B' are formed by rotating vectors A and B:
$$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$
$$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...
Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS:
##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'##
I posted this question long ago and i was told that this is the scalar product...
Homework Statement
How does the scalar product of displacement four vector with itself give the square of the distance between them?
Homework Equations
(Δs)2= Δx.Δx ( s∈ distance, x∈ displacement four vector)
or how
ds2=ηαβdxαdxβ
The Attempt at a Solution
Clearly I am completely new to the...
Homework Statement
If at some particular place and time the sun light is incident on the surface of the Earth along
a direction defined by the unitary vector – vˆ , where vˆ =(4, 3, 5)/sqrt (50) and with a power
density P, what is the total power captured by a solar panel of 1.4 m2
and with...
Homework Statement
Find angle between vectors if
\cos\alpha=-\frac{\sqrt{3}}{2} [/B]Homework EquationsThe Attempt at a Solution
Because cosine is negative I think that \alpha=\frac{5\pi}{6}. But also it could be angle \alpha=\frac{7\pi}{6}. Right? When I search angle between vectors I do not...
Vectors a and b correspond to the vectors from the origin to the points A with co-ordinates (3,4,0) and B with co-ordinates (α,4, 2) respectively. Find a value of α that makes the scalar product a\cdotb equal to zero, and explain the physical significance.
My attempt:
The scalar product...
The Vector A points 17° counterclockwise from the positive x axis. Vector B lues in the first cuadrant of the xy plane. The magnitudes of the cross product and the dot product are the same:
i.e, |AXB|= |A(times)B|
What Angle does B make with the positive x axis?
2. Is ti a scalar...
Homework Statement
For what values of k is (scalar product of vectors a and b) = a_{1}b_{1}-a_{1}b_{2}-a_{2}b_{1}+ka_{2}b_{2} a valid scalar product?
The vectors a and b are defined as:
a = a_{1}e_{1} + a_{2}e_{2}
b = b_{1}e_{1} + b_{2}e_{2}
where e_{1} and e_{2} are unit vectors...
Hi,
can somebody help me with the problem:
Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows...
Homework Statement
Hi all,
Here's the problem:
Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C.
Homework Equations
The Attempt at a Solution
I started by looking at the triple...
Hello,
I am confused how vectors that are coplanar will give a triple product of zero? Or is it the case that all 3 vectors must be coplanar for a triple product of zero, or is 2 sufficient? I.e. the vector being dotted with one of the vectors being crossed in the same plane, will this...
Homework Statement
Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A)
Show that this indeed fulfils the requirements on a scalar product.
Homework Equations
Conditions for a scalar...
Homework Statement
Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that
defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product?
Homework Equations
The Attempt at a...
Im having trouble understanding this property
my book states that: a.(bxc) = b.(cxa) = c.(axb)
it also states that a.(ax(anything)) = 0
I understand the second point and why that's true, what I don't understand is why a.(bxc) = b.(cxa) = c.(axb) is true
If I name any 3 vectors a b...
Homework Statement
Given the vectors:
P = 8i +5j-Pzk m
and
Q = 3i -4j-2k m
Determine the value of Pz so that the scalar product of the two vectors will be 60m2Homework Equations
Sure seems like we will need to use the following equation:
P * Q = |P| * |Q| * cos ∅
But I don't recall being able...
Homework Statement
Reading a textbook, I come across a situation where an electron is scattered off a nucleus. The book says p.P = p'.P', where p is the momentum of the electron and P is the momentum of the nucleus.
I don't understand how it gets the conservation of scalar product.
It's steps...
Homework Statement
I'm confused about the difference between the following two statements:
\mathbf{V_1}\mathbf{V_2}=V_1V_2\cosh (\phi)
and
\mathbf{V_1}\mathbf{V_2}=\gamma c^2
Where \gamma is the Lorentz factor of the relative speed between the two vectors. Both vectors are time-like.
The...
Homework Statement
If M is a square matrix, prove:
(A, MB) = (adj(M)A, B)
where (A, MB) denotes the scalar product of the matrices and adj() is the adjoint (hermitian adjoint, transpose of complex conjugate, M-dagger, whatever you want to call it!)
Homework Equations
adj(M)=M(transpose of...
Hello!
I seem to have a problem with spherical coordinates (they don't like me sadly) and I will try to explain it here. I need to calculate a scalar product of two vectors \vec{x},\vec{y} from real 3d Euclidean space.
If we make the standard coordinate change to spherical coordinates we can...