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.)
Homework Statement
If ##u,v,w\in\mathbb{R}^3##, show that ## u\times(v\times w) = (u.w) v - (u.v) w ##.
Homework Equations
The Attempt at a Solution
Since ## u\times(v\times w)##, ##v## and ##w## are orthogonal to ##v\times w##, these vectors are coplanar. Therefore, there must be reals ##...
Homework Statement
Show that the product of two nxn unitary matrices is unitary. Is the same true of the sum of two nxn unitary matrices?
Homework Equations
Unitary if A†A=I
Where † = hermitian conjugate
I = identity matrix.
The Attempt at a Solution
[/B]
We have the condition: (AB)†(AB)=I
I...
Reading a book about 3d math, and I am confused as to what happened on this Vector Cross Product problem. I'm thinking there was just an error that wasn't caught.
For the first row, instead of (3)(8)-(-4)(-5) shouldn't it have been (3)(8)-(4)(-5) and had the same displayed result of 44?
And for...
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
Problem. Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules.
If $M\otimes_R N=0$, then $M=0$ or $N=0$.
The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where...
Homework Statement
Hi, I wasn't sure whether to post this here or in the engineering forums. Since it's mainly math/theory I figured here would be more appropriate. Feel free to move it if it doesn't belong here. All relevant info etc. is in the picture, thanks.
Homework Equations
The...
Homework Statement
Given basis |x>,|y>,|z> such that <x|x> = 2,<y|y> = 2,<z|z> = 3,<x|y> = i, <x|z> = i, and <y|z> = 2. Build an orthonormal basis|x'>,|y'>,|z'>. Each of the new basis vectors should be expressed in terms of the old ones multiplied by coefficients.
Homework Equations
|x'> =...
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...
Here is a mystery I'm trying to understand. Let ##\hat{U}(t) = \exp[-i\hat{H}t]## is an evolution operator (propagator) in atomic units (\hbar=1). I think I'm not crazy assuming that ##\hat{U}(-t)\hat{U}(t)=\hat{I}## (unit operator). Then I would think that the following should hold
\left\langle...
Homework Statement
[/B]
Vector A lies in the yz plane 63.0 degrees from the +y axis, has a positive z component, and has a magnitude 3.20 units. Vector B lies in the xz 48.0 degrees from the +x axis, has positive z component, and has magnitude 1.40 units.
a) find A dot B
b) find A x B
c)...
Hello, PF!
I had a quick question that I hoped maybe some of you could help me answer. The question is simple: Why is the cross product of two parallel vectors equal to the zero vector? I can see this easily mathematically through completing the cross product formula with two parallel...
The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product.
I want to show that:
##det A \overset{!}{=} a_1 \cdot (a_2 \times...
It seems there should be a list of tensor identities on the internet that answers the following, but I can't find one.
For tensors in ##R^4##,
##S = S_\mu{}^\nu = S_{(\mu}{}^{\nu)}## is a symmetric tensor.
##A = A_{\nu\rho\sigma}= A_{[\nu\rho\sigma]}## is an antisymmetric tensor in all...
Let's say we have:
\vec{E}=E_x\vec{i}_x+E_y\vec{i}_y+E_z\vec{i}_z
and
\vec{B}=B_x\vec{i}_x+B_y\vec{i}_y+B_z\vec{i}_z
and the Lorentz Force
0=q(\vec{E}+\vec{v}X\vec{B})
which due to
\vec{E}X\vec{B}=\vec{B}X(\vec{v}X\vec{B})=vB^2-B(\vec{v}\cdot \vec{B})
and transverse components only...
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:
$$A'_i=\sum_j \lambda_{ij} A_j,\; B'_i=\sum_j \lambda_{ij} B_j$$
$$A' \centerdot B'=\sum_i A'_i B'_i =\sum_i \left( \sum_j \lambda_{ij} A_j \right)\left( \sum_k...
Homework Statement
https://www.dropbox.com/s/8l90hahznjlv9d0/vector%20problem.png?dl=0
Homework Equations
Dot and Cross product
The Attempt at a Solution
although I know the dot and cross product, I'm not sure what I'm being asked or how to proceed? any help?[/B]
Hi all.
I'm struggling with taking dot products between vectors in spherical coordinates. I just cannot figure out how to take the dot product between two arbitrary spherical-coordinate vectors ##\bf{v_1}## centered in ##(r_1,\theta_1,\phi_1)## and ##\bf{v_2}## centered in...
Suppose we have the sets $A=\left\{2,3\right\}$ and $B=\left\{5\right\}$, then $A$ X $B$ is defined as $\left\{(x,y)|x \in A, y\in B\right\}=\left\{(2,5), (3,5)\right\}$. But what happens when $A$ contains elements that are not in $\Bbb{R}$?
Example:
$A=\left\{(2,3),(3,4)\right\}\subset...
Homework Statement
Prove that $$log_{b}(xy)=log_{b}x+log_{b}y.$$
Homework Equations
Let $$b^{u}=x,b^{v}=y.$$ Then $$log_{b}x=u,log_{b}y=v.$$
The Attempt at a Solution
I'm afraid I've been using circular reasoning to prove this. I can get this to a point where I have...
Two lines A and B. The angle between them is θ, their direction cosines are (α,β,γ) and (α',β',γ'). Prove, ON GEOMETRIC CONSIDERATIONS:
##\cos\theta=\cos\alpha\cos\alpha'+\cos\beta\cos\beta'+\cos\gamma\cos\gamma'##
I posted this question long ago and i was told that this is the scalar product...
Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.
Hello everyone, I would like to know if anyone knows what is the inner product for vector fields ##A_\mu## in curved space-time. Is it just:
$$
(A_\mu,A_\mu)=\int d^4x A_\mu A^\mu =\int d^4x g^{\mu\nu}A_\mu A_\nu
$$
? Do I need extra factors of the metric?
Thanks!
How does one show that
eAeB=eA+Be[A,B]/2
where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...
How does one show that
eAeB=eA+Be[A,B]/2
where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...
Homework Statement
Using the equations that are defined in the 'relevant equations' box, show that
$$\langle n' | X | n \rangle = \left ( \frac{\hbar}{2m \omega} \right )^{1/2} [ \delta_{n', n+1} (n+1)^{1/2} + \delta_{n',n-1}n^{1/2}]$$
Homework Equations
$$\psi_n(x) = \left ( \frac{m...
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
Let G = G1 × G2 be the direct product of two simple groups. Prove that every normal subgroup of G is isomorphic to G, G1, G2, or the trivial subgroup.
The Attempt at a Solution
I tried proving that the normal subgroups would have to be of the form Normal subgroup X Normal...
Say I have a position vector
p = e(t) p(t)
Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T
And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T
I want the velocity, so I take
v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...
Hello, I hope this is the right forum section.
I'm having trouble understanding how calculating the cross product arrives at the final result. When I do something simpler like multiplying a vector by a scalar, I can easily visualize in my head how each component "shrinks" or "grows".
With the...
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...
When do functions have representations as a "direct product"?
For example, If I have a function f(x) given by the ordered pairs:
\{(1,6),(2,4),(3,5),(4,2),(5,3),(6,1) \}
We could (arbitrarily) declare that integers in certain sets have certain "properties":
\{ 1,3\} have property A...
Homework Statement
Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.)
Homework Equations...
Hi,
I'm having some issues with a piece of my notes. (relevant pages attached)
First we introduce an isomorphism ##U = \oplus_n U_n## from ##\Gamma^{(a)s}\left(\mathcal{H}_1\oplus\mathcal{H}_2\right)## to ##\Gamma^{(a)s}\left(\mathcal{H}_1\right)\otimes\Gamma^{(a)s}\left(\mathcal{H}_2\right)##...
Homework Statement
Vectors A and B both have magnitude M. Joined at the tails, they create a 30' angle. What is A x B in terms of M?
Homework EquationsThe Attempt at a Solution
0? OR M^2? Sqrt(3)M/3?
So the following question is attached (There is another thread with the same question but no solution to what I am asking on there)
Now according to several solutions, apparently IYZ is equal to 0, and they reason this by saying that the geometry is symmetrical.
However when looking at the...
Hi their,
It's a group theory question .. it's known that
## 10 \otimes 5^* = 45 \oplus 5, ##
Make the direct product by components:
##[ (1,1)^{ab}_{1} \oplus (3,2)^{ib}_{1/6} \oplus (3^*,1)^{ij}_{-2/3} ] \otimes [ (1,2)_{ c~-1/2} \oplus (3^*,1)_{ k~1/3} ] = (1,2)^{ab}_{ c~1/2} \oplus...
I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...
This is the gravitational potential energy formula
$$U = -\int_\infty^r\vec{F}_\text{field}\cdot d\vec{r}$$
If r vector's direction is form infinity to r, then it means it has same direction as Gravitational Force. So cos0=1
But after multiplication there is a negative sign here: "-GMm"
$$U =...
Pre-knowledge
A matrix is a linear transformation if, T(u+v)= T(u) +T(v) and T(cu)=cT(u).
Theorem 8.4.2 If V is a finnite dimensional vector space, and T: V-> V is a linear operator then the following are equivalent.
a) T is one to one, b) ker(T)=0, c)...
Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete).
Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X.
Relevant equations:
S^{\perp} is always...
Hello Physics Peeps,
It just came up in the notes for my electrodynamics class that an electrons charge squared can be expressed as the radius times the mass times the speed of light squared.
e^2 = m_er_ec^2
I don't understand the motivation for doing this. I've tried to search for other...
Let \mathbb{F} be an arbitrary field, and let a,b\in\mathbb{F}^3 be vectors of the three dimensional vector space. How do you prove that if a\times b=0, then a and b are linearly dependent?
Consider the following attempt at a counter example: In \mathbb{R}^3
\left(\begin{array}{c}
1 \\ 4 \\ 2...