Cross product Definition and 471 Threads

why force parallel to the axis of rotation do not make any torque.

I understand that the torque on a gyrating object is defined as the force vector cross multiplied by the lever arm position vector, which produces a resultant vector that is normal to both of the original vectors. However, when an object (let's say a disk) is rotating about an axis...

The Lorentz transformation operator acting on an undotted, i.e. right-handed, spinor can be expressed as $$e^{-\frac{1}{2} \sigma \cdot \mathbf{\phi} + i\frac{1}{2} \sigma \cdot \mathbf{\theta}}.$$ There is a very cool, almost childlike, derivation of this expression in Landau Vol. 4 S. 18 I've...

Homework Statement Given the figure below[/B] I need to calculate the total magnetic force on the semicircle section of the conductor. Current is I, Radius is R, and the Magnetic Field is B. Homework Equations d\vec{F} = Id\vec{l} \times \vec{B}[/B] The Attempt at a Solution [/B] dl is...

Mod note: Reproduced contents of image with broken link: i = j x k j = k x i k = i x j Wikipedia says this about the standard basis vectors. Does this work for all (i.e, non basis) vectors? For example, if you know A = B X C does that mean C = A X B and B = C X A?

Homework Statement This is not a homework problem, I am currently reading the Derivation of potential of a charged particle in Electric and Magnetic field from the book Mechanics by Symon (I attached the image of the page), I need to know how to expand the vector cross product such as...

Homework Statement Suppose r:R\rightarrow { V }_{ 3 } is a twice-differentiable curve with central acceleration, that is, \ddot { r } is parallel with r. a. Prove N=r\times \dot { r } is constant b. Assuming N\neq 0, prove that r lies in the plane through the origin with normal N. Homework...

The first part of the problem was easy enough as was finding the directions of the other two, but I am having trouble finding the correct angle for the last two cross products in order to find the magnitude.

Homework Statement The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|. Homework EquationsThe Attempt at a Solution I know that the magnitude of the cross product of any two vectors...

Homework Statement A\cdot B\times C\quad =\quad 2\\ (2A+B)\quad \cdot \quad [(A-C)\quad \times \quad (2B+C)]\quad =\quad ? Homework Equations Various cross product and dot product properties The Attempt at a Solution I've only managed to get so far, don't really know what to do next A\cdot...

I am trying to derive the law of signs from the cross product. First, we have three vectors ##\vec{A} ~\vec{B} ~\vec{C}## such that ##\vec{A} + \vec{B} + \vec{C} = 0##. This creates a triangle. Then, we label the angles opposite the respective sides as a, b, and c. I am not sure where to go...

If we seek a bijection $$\wedge^p V \to \wedge^{n-p} V$$ for some inner product space ##V##, we might think of starting with the unit ##n##-vector and removing dimensions associated with the original vector in ##\wedge^p V ##. Might this be expressed as a sequence of steps by some binary...

Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.

The determinant of some rank 2 tensor can be expressed via the exterior product. $$T = \sum \mathbf{v}_i \otimes \mathbf{e}_i \;\;\; \text{or}\sum \mathbf{v}_i \otimes \mathbf{e}^T_i $$ $$ \mathbf{v}_1\wedge \dots \wedge \mathbf{v}_N = det(T) \;\mathbf{e}_1\wedge \dots \wedge\mathbf{e}_N$$ The...

Homework Statement Using index-comma notation only, show: \begin{equation*} \underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} \end{equation*} Homework Equations...

Homework Statement I was checking my work and Chegg uses the equations differently. Can somebody tell me why? Maybe I'm misunderstanding how/why to use the equation I chose. Homework Equations They say aB = -ω2ABRB/Ai I used aB = aA + αk x r - ω2rB/A The Attempt at a Solution So obviously...

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...

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...

In the cross product, why is vectorA*B=-(vectorB*A) How does the right-hand rule apply to this formula?

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>...

I'm not 100% confident of my approach to the 2 exercises below: Orbital angular momentum of i'th element is $\vec{L_i} = \vec{r_i} \times \vec{p_i} = m_i \vec{r_i} \times (\omega \times \vec{r_i}) $ a) Find the inertia matrix $I$ such that (omitting vector signs from here on) $L = I \omega, |L...

Homework Statement A rotational axis is directed perpendicular to the plane of a square and is located as shown in the drawing. Two forces, F1 and F2, are applied to diagonally opposite corners, and act along the sides of the square, first as shown in part a and then as shown in part b of the...

If the cross product in ℝ3 is defined as the area of the parallelogram determined by the constituent vectors joined at the tail, how does one go about proving this product to distribute over vector addition? I've attached a drawing showing cyan x yellow, cyan x magenta, and cyan x (magenta +...

So its been a while since I studied maxwells equations, anyway: So From my ignorant perspective, trying to derive conceptual meaning from these, I can see that the time dependant study there is some conductivity x the partial differential of the magnetic vector potential plus the cross product...

Hi - from orbital angular momentum components, $[L_x, L_y] = iL_z$ My book claims 'Hence, $ \vec{L} \times \vec{L} = i\vec{L} $' I'm keen to know how they get that, an also why that cross products isn't = 0, like $A \times A$ would be ?

I got to here in a simple exercise (orb. ang. momentum cords), realized I was applying something I didn't understand ... $L = -i \begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\x&y&z\\\pd{}{x}&\pd{}{y}&\pd{}{z}\end{vmatrix}$ I 'know' it equates to $L_x =-i \left( y\pd{}{z} - z\pd{}{y} \right) $ - but...

Homework Statement Angular momentum is the cross product of r and mv. But why is there mvR outside of the paranthesis? And where did the v go in the second paranthesis - shouldn't the second paranthesis be (-v*sin(ωt), v* cos(ωt)). Does anyone have any idea how they did the cross product...

Lets look at the force on a wire segment in a uniform magnetic field F = I∫(dl×B) I am curious if, from this, we can say: F = I [ (∫dl) × B] since B is constant in magnitude and direction

A method for finding the shortest distance between 2 skew, non intersecting lines is to 1st find the common normal, using $ \vec{n} = \frac{\vec{v_1} \times \vec{v_2}}{|\vec{v_1} \times \vec{v_2}|} $ I'm looking for a proof or intuition as to why this is true please? Then apparently we get the...

Hi, I was just wondering if you have a cross product can you multiply out the constants and put them to one side. So ik x ik x E is equal to i^2(k x k x E) therefore is equal to -k x k x E. Is that correct?

So yeah, I understand that you can calculate torque as F*d, and you get a "number". But when you calculate a cross product of torque, r x F, what does that actually give you? It is a vector, perpendicular to F and r, but what "is" that? I mean, is it like an axis around which the object is...

Homework Statement The problem is given in the following photo: Actually I did the first proof but I couldn't get the second relation. (Divergence of E cross H). Homework Equations They are all given in the photo. (a) (b) and (c). The Attempt at a Solution What I tried is to interchange...

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 ##...

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 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...

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...

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...

Homework Statement A uniform cylinder of mass M and radius R can be rotated about a perpendicular axle through its centre. A particle of mass m is attached to the cylinder's rim. The system is rotated with angular velocity w about the axle, which is held in a fixed direction during the motion...

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...

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]

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 I need to proof that VxU=(determinant) starting from VxU=|V||U|sinαe Homework Equations VxU=|V||U|sinαe and what I'm aiming to is VxU=(uy⋅uz - uz⋅vy)i - (ux⋅vz - uz⋅vx)j + (ux⋅vy - uy⋅vx)k The Attempt at a Solution U x V = |U||V|Sinαe (U x V)^2 = |U|^2|V|^2 cos^2α - 1 e (U...

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?

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 \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...

Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...

Can anyone tell me why?

Homework Statement Given a×b=-i-j+3k and c×a=2i-3j+k, find a×(a-2b+c) Homework Equations Cross product (DONE WITHOUT MATRICES). The Attempt at a Solution a[/B]×b=c=-(b×a)is all I'm getting to at this point

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...