In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space
R
3
{\displaystyle \mathbb {R} ^{3}}
, and is denoted by the symbol
×
{\displaystyle \times }
. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).
If two vectors have the same direction or have the exact opposite direction from one another (i.e., they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths.
The cross product is anticommutative (i.e., a × b = − b × a) and is distributive over addition (i.e., a × (b + c) = a × b + a × c). The space
R
3
{\displaystyle \mathbb {R} ^{3}}
together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to a pseudovector, or the exterior product of vectors can be used in arbitrary dimensions with a bivector or 2-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can, in n dimensions, take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. (See § Generalizations, below, for other dimensions.)
Hello
I found a bug in my code and can't figuring out the error. I tried debugging by showing the output of each variable step by step but I can't find my error. Here is what I have and what I want to do:
I have a matrix A:
0000
0101
1010
1111
And I have a matrix B:
10000
21000
30100
41100...
(Not an assigned problem...)
1. Homework Statement
pg 244 of "Mathematical Methods for Physics and Engineering" by Riley and Hobson says that given the following two properties of the inner product
It follows that:
2. Attempt at a solution.
I think that both of these solutions are...
1. Homework Statement
For part a), I know that it is a nucleophilic addition reaction of CN- to form cynohydrin, but how come there are 2N atoms in E? How does NH4+ react with CH3CHO or the cynohydrin formed?
For part b), what reaction is this?
Homework EquationsThe Attempt at a Solution
I recently got confused about Lie group products.
Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...
Homework Statement
A point P moves so that its distances from A(a, 0), A'(-a, 0), B(b, 0) B'(-b, 0) are related by the equation AP.PA'=BP.PB'. Show that the locus of P is a hyperbola and find the equations of its asymptotes.
Homework EquationsThe Attempt at a Solution
AP.PA' =...
1. Homework Statement
Why is the product at anode Br2 for MgBr2? Instead of O2?
Homework EquationsThe Attempt at a Solution
[/B]
The EΘ value for the oxidation of OH- is -0.40 V,
And that of Br- is -1.07V
And so, shouldn't oxidation of OH- be easier? And hence, O2 is formed at the anode?
Is there any proof for the matrixx formula of the cross product. I am asking this because I have seen many videos and they have used the matrixx formula and then proved that ||A X B|| = ||A|||B||sin(theta), khan academy also used the same method
Homework Statement
Hi everyone,
I am a first year physics student and we recently learned about torque.
Every time I think I understand it something else comes up to confuse me - this time it is the direction. I tried looking in the forum and generally in google, but everyone only explains the...
(Scalar)·(Scalar) = Scalar
(Scalar)·(Vector) = Scalar
(Vector)·(Vector) = Scalar
(Scalar)x(Scalar) = Not valid
(Scalar)x(Vector) = Vector
(Vector)x(Vector) = VectorDid I get them right, if not why?
Thanks
They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices.
1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$
= $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I...
What is meant by taking the tensor product of vectors? Taking the tensor product of two tensors is straightforward, but I am currently reading a book where the author is talking about tensor product on tensors then in the next paragraph declares that tensors can then be constructed by taking...
What does the angle theta acutally means in cross product because I have seen in many places it is written that theta is the angle at which two vector on a given plane will coinside with each other so that there will be only one direction. Is it true and why they defined it in this way , I...
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...
If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...
I know that ##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu} \phi \big) = \partial_{\mu} \phi##.
Now, I need to prove this to myself.
So, here goes nothing.
##\frac{\partial}{\partial (\partial_{\mu}\phi)} \big( \partial_{\mu} \phi\ \partial^{\mu}...
A particle in a 1-D Hilbert space would have position basis states ## |x \rangle ## where ## \langle x' | x \rangle = \delta(x'-x) ## A 3-D Hilbert space for one particle might have a basis ## | x,y,z \rangle ## where ##\langle x', y', z' | x,y,z \rangle = \delta(x'-x) \delta (y-y') \delta(z-z')...
Homework Statement
I am trying to determine whether
$$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$
where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions.
Homework Equations
The defining equation of a delta function...
Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...
I've been working on a problem for a couple of days now and I wanted to see if anyone here had an idea whether this was already proven or where I could find some guidance. I feel this problem is connected to the multinomial theorem but the multinomial theorem is not really what I need . Perhaps...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows:
I do not...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure 6.1 which is...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
About midway in the above text, just at the start...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:https://www.physicsforums.com/attachments/5391...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows:
In the above text...
For any $a \in \mathbb{R}$, let $a^3$ denote $a \cdot a \cdot a$. Let $x, y \in \mathbb{R}$.
1. Prove that if $x < y$ then $x^3 < y^3$.
2. Prove that there are $c, d \in \mathbb{R}$ such that $c^3 < x < d^3$.
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an...
Hi all got a confusion
In many books I saw , authors used a specific statement here is it
a,b,c are vectors and axb is (" a cross b")
In general
(axb)xc ≠ ax(bxc)
but if
(axb)xc = ax(bxc)
solving it we get
bx(axc)=0
then it implies
either b is parallel to (axc)
or a and c are collinear...
Homework Statement
e) If H ∼= Z3 × Z3 show that there are exactly 2 conjugacy classes of elements of order 2 in Aut(Z3 × Z3) = GL(2, Z3).
f) Choosing an element of each conjugacy class in e), construct two semidirect products of H and K. By counting orders of elements in each such group, show...
Hello! (Wave)
Does it suffice to show that the triple product is 0?
If we show that $a \cdot (b \times c)=0$ we will have that $a$ is orthogonal to $b \times c$. $b \times c$ is orthogonal to both $b$ and $c$, so we will have that $a$ will be parallel to $b$ and $c$.
Right? But why does...
I am reading John M. Lee's book: Introduction to Smooth Manifolds ...
I am focused on Chapter 3: Tangent Vectors ...
I need some help in fully understanding Lee's definition and conversation on pushforwards of F at p ... ... (see Lee's conversation/discussion posted below ... ... )
Although...
I am trying to show that if (C^ab)(A_a)(B_b) is a scalar for arbitrary vectors A_a and B_b then C^ab is a tensor.
I want to take the product of the two vectors then use the quotient rule to show that C^ab must then be a tensor. This lead to the question of whether or a not the product of two...
Hello,
if we consider a group G and two subgroups H,K such that HK \cong H \times K, then it is possible to prove that:
G/(H\times K) \cong (G/H)/K
Can we generalize the above equation to the case where HK \cong H \rtimes K is the semidirect product of H and K?
Clearly, if HK is a semidirect...
Hi all,
The basis vectors are defined as 1x3 matrices, how can the result be a 3x3 matrix?
How can the result of a dot product be a 3x3 matrix, I'm stumbled, how can I evaluate this?
A inner product returns a scalar, and now it returns a 3x3 matrix, please help.
Thanks.
We increase by 1 each of three prime numbers, not necessarily distinct. Then we
form the product of these three sums. How many numbers between 1999 to 2021
can appear as such a product?
Hi,
The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation
Vm= (- 1/B2 ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A2 ...[1]
the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m>...
Hi,
In my QFT course, the professor writes an infinite product like this:
∏n | k n0 > 0 ∫...
My question is, what does the `|' in the subscript "n | k" representing? When I see `|', I think logical OR - obviously that is not it. Normally, if it's a sum over two indices, commas separate the...
Homework Statement
Find the gradient of \underline{\nabla}(\underline{a}\cdot\underline{r})^n where a is a constant vector, using suffix notation and chain rule.
Homework Equations
On the previous problem,s I found that grad(a.r)=a and grad(r)=\underline{\hat{r}}
The Attempt at a Solution...
I would like to gain a more formal mathematical understanding of a construct relating to spinors.
When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work...