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: About semidirect product of Lie algebra
Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E##
Hi,
Please, I have a question about the module of special lie algebra:
Let ##\mathbb{K}## be a field. Let the Lie algebra...
For each positive integer ##m##, let ##C_m## denote a cyclic group of order ##m##. Show that for all positive integers ##m## and ##n##, there is an isomorphism ##C_m \times C_n \simeq C_d \times C_l## where ##d = \operatorname{gcd}(m,n)## and ##l = \operatorname{lcm}[m,n]##.
An infinite product representation of Bessel's function of the first kind is:
$$J_\alpha(z) =\frac{(z/2)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^\infty(1-\frac{z^2}{j_{n,\alpha}^2})$$
Here, the ##j_{n,\alpha}## are the various roots of the Bessel functions of the first kind. I found this...
So far, I have got the equations,
##u \cdot (\vec u \times \vec v) = 0##
##u_1a + u_2b + u_3c = 0##
##v_1a + v_2b + v_3c = 0##
Could some please give me some guidance?
Many thanks!
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
Hi. I hope everyone is well. I'm just an old person struggling to make sense of something I've read and I would be very grateful for some assistance. This is one of my first posts and I'm not sure all the LaTeX encoding is working, sorry. Your help pages suggested I add as much detail as...
Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
How is the mass inertia product calculated? I have two examples and each one uses something different.
Example 1:
Example 2: moments and product of inertia of the cylinder
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...
We have three Random variable or vector A,B,C. Condition is A & B are independent as well as B & C are independent RVs . But A & C are the same random variable with same distribution . So How can determine E{ABC}. Can I write this E{ABC}= E{AE{B}C}?
If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
\lim_{x \to \infty}f(x)g(x)=0
I found that only for sequences, but it should...
Hi!
For example, how do you tell whether to use the scalar or cross product for an problem such as,
However, I do know that instantaneous angular momentum = cross product of the instantaneous position vector and instantaneous momentum. However, what about if I didn't know whether I'm meant to...
Hi!
For this problem,
The solution is,
However, I don't understand their solution at all. Can somebody please explain their reasoning in more detail.
Many thanks!
Could anyone explain the reasoning from step 2 to step 3?
Specifically, I don't understand how to find the product of a cross product and a vector - like (v1 · v2)v1 and (v1 · v3)v1. I'm also confused by v1 × v3 + (v1 · v3)v1 -- is v1 × v3 = v1v3? How would this be added to (v1 · v3)v1?
Thank you.
Hi Pfs
i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u>
If i take their tensor product i will get 4*4 matrices with this basis:
d>d>,d>u>,u>d>,u>u>
these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...
Hi everyone
I have the solutions for the problem. It makes sense except for one particular step.
Why does the dot product of a and b equal zero? I thought this would only be the case if a and b were at right angles to each other. The solutions seem to be a general proof and should work for...
The proof is by strong induction.
Suppose ## p ## is an S-prime.
Then ## p=4k+1 ## for some ## k\in\mathbb{N} ##.
Let ## n ## be an S-composite such that ## n=p_{1}^{k_{1}}p_{2}^{k_{2}}\dotsb p_{r}^{k_{r}} ## where ## p_{i} ## are all S-primes.
(1) When ## k=1 ##, the statement is ## p=4(1)+1=5...
Given an orthonormal basis ##\{e_1,\ldots, e_n\}## in a complex inner product space ##V## of dimension ##n##, show that if ##v_1,\ldots, v_n\in V## such that ##\sum_{j = 1}^n \|v_j\|^2 < 1##, then ##\{v_1 + e_1,\ldots, v_n + e_n\}## is a basis for ##V##.
The magnitude of cross product is defined of vector A⃗ and B⃗ as |A⃗×B⃗|=|A⃗||B⃗|sinθ where θ is defined as the angle between the two vector and 0≤θ≤π.the domain of θ is defined 0≤θ≤π so that the value of sinθ remains positive and thus the value of the magnitude |A⃗||B⃗|sinθ also remain positive...
Hi
In QM the inner product satisfies < a | a > ≥ 0 with equality if and only if a = 0.
Is this positive definite or positive semidefinite because i have seen it described as both
Thanks
Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$
where $v_1,w_1...
##\langle T(f), g \rangle = \int_{0}^{1} \int_{0}^{x} f(t) dt ~ g(t) dt##
As ##\int_{0}^{x} f(t) dt## will be a function in ##x##, therefore a constant w.r.t. ##dt##, we have
##\langle T(f), g \rangle = \int_{0}^{x} f(t) dt ~ \int_{0}^{1} g(t) dt##
##\langle f, T(g)\rangle = \int_{0}^{1} f(t)...
Hello
Simple question
Whether the minimum of the product of two functions in one single variable, is it greater or less than the product of their minimum
thanks
Sarrah
I am refreshing on this...Have to read broadly...i will start with (b) then i may be interested in alternative approach or any correction that may arise from my working. Cheers.
Kindly note that i do not have the solutions to the following questions...
For (b), we know that,
say, if ##x=α##...
Hello guys, I'm a newbie.
So I have developped an application that rotates a cube using quaternion.
The initial values of the quaternion are ( w=1.0, x=0.0, y=0.0, z=0.0).
Now I want to apply two consecutive rotation using two different quaternion values:
The first rotation corresponds to...
We mainly have to prove that this quantity
## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ##
is greater or equal than zero for all ##\ket{\varphi}##.
Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I...
We can visualize 1-form by contour lines, since a 1-form / gradient sort of represents how fast the function changes. I wonder whether we can visualize 2-form df ^ dg by intersection of two sets of contour lines for f and g, or maybe something of a similar nature?
Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...
Hi,
I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea.
Question:
Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve...
Proof:
Suppose ## n>1 ## is a positive integer.
Let ## n=p_{1}^{k_{1}} p_{2}^{k_{2}}\dotsb p_{r}^{k_{r}} ## be the prime factorization of ## n ##
such that each ## k_{i} ## is a positive integer and ## p_{i}'s ## are prime for ## i=1,2,3,...,r ## with
## p_{1}<p_{2}<p_{3}<\dotsb <p_{r} ##...
Let ##\omega## be 2-form and ##\tau## 1-form on ##R^3## If X,Y,Z are vector fields on a manifold,find a formula for ##(\omega\bigwedge\tau)(X,Y,Z)## in terms of the values of ##\omega## and ##\tau ## on the vector fields X,Y,Z.
I have known how to deal with only one vector field.But there are...
Summary:: summation of the components of a complex vector
Hi,
In my textbook I have
##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}##
##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}##
For ##\hat{e_p} = \hat{x}##...
We've two operators ##\hat{a}##,##\hat{b}##. I know their position representation ##\langle r|\hat{b} \mid \psi\rangle=b##
##\langle r|\hat{a}| \psi\rangle=a
##
Is it generally true that the position representation of the combined operator ##\hat{a}\hat{b}## is ##a b## where ##a, b## are the...
y = 10*(1 + cos(0.1*x)) --> dy/dx = -sin(0.1x)
dW = F*dx + F*dy = 10*sin(0.1*x)dx + 10*sin(0.1*x)*-sin(0.1x)
integrating we have -100*cos(0.1*x) -10*sin(0.1x)^2 from 0 to 10*pi = W = 43 J. The answer says 257 J. Where am I wrong here?
I am not sure what I am doing wrong but dot product of a and b =/= |a||b| when I am trying to calculate it. Theta = 0:
dot product(a and b) = ax*bx + ay*by
|a||b|= sqrt((ax^2+ay^2)*(ax^2 + by^2)) = sqrt((ax*bx)^2 + (ax*by)^2 + (ay*bx)^2 + (ay*by)^2) =/= ax*bx + ay*by
What am I doing wrong?
I'm trying to calculate the electrostatic energy, and I'm wondering what happens when I dot the D-field and E-field, with Si-units V/m**2. This is my equation:
D dot E = (-4x(epsilon) V/m**2)(-4x V/m**2) + (-12y(epsilon) V/m**2)(-12y V/m**2)
Are the final Si-unit still V/m**2 or V**2/m**4?
For part (i),
##(x-α)(x-β)=x^2-(α+β)x+αβ##
##α+β = p## and ##αβ=-c##
therefore,##α^3+β^3=(α+β)^3-3αβ(α+β)##
=##p^3+3cp##
=##p(p^2+3c)##
For part (ii),
We know that; ##tan^{-1} x+tan^{-1} y##=##tan^{-1}\left[\dfrac...
For part a,
We have ##α+β=b## and ##αβ =c##. It follows that,
##(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)##
=##α^2β^2+(α+β)^2-2αβ +1##
=##c^2+b^2-2c+1##
=##c^2-2c+1+b^2##...
My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question.
I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...
When I look at a range of inputs around x=c and consider the corresponding range of outputs
If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively.
|f(x)-L1||g(x)-L2|<ϵ2ϵ1
The product of the...
In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,
T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ]
Since Einstein equation is G = 8πGT, I...
I am sure you are all familiar with the cross product in 3D space.
i cross into j gives k.
Cyclic
Negative, if reversed, etc.
I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...
Using the inequality of arithmetic and geometric means,
$$\frac {x+y}{2}≥\sqrt{xy}$$
$$6^2≥xy$$
$$36≥xy$$
I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.