Cross product Definition and 465 Threads

If A\in O(3) and v,w\in\mathbb{R}^3, then (Av)\times (Aw) = (\textrm{det}\;A) A(v\times w) is right, right? Is there any simple way of proving this? It doesn't come easily like invariance of dot product. Isn't this equation also important for understanding the pseudovectors?

Homework Statement I need to calculate Ax(BxC) -A, B, C are vectors, apart from bac-cab rule, I need to get it by epsilon ijk -Following I will note it by e(ijk). Homework Equations I expect answerer to know kronecker delta equation and vector multiplications. The Attempt at a...

How do you prove that ||VxU|| is the area of the parallelegram they form? I know that the cross product is a vector perpendicular to V and U

I am reading through David Widder's Advanced Calculus and he abbreviates a determinant as: \left( \begin{array}{cccc} r_{1} \ s_{1} \ t_{1}\\ r_{2} \ s_{2} \ t_{2}\\ r_{3} \ s_{3} \ t_{3}\\ \end{array} \right) And refers to it by (rst). He then states...

Homework Statement I have came up with an example to illustrate my question. There is a rod, which can turn around p1. p1p2 = (-1+j) m p1p3 = (-3 + 3j) m p1p4 = (1 - j ) m F1 = (1+3j) N F3 = (-1 - 2j ) N F4 = unknown, orthogonal to the rod compute F2_n, orthogonal component of F2 to the...

Homework Statement C= B|A| + A|B| D= A|B|-B|A| C and D are orthogonal Find a third vector perpendicular to both C and D Homework Equations [AxB] = |A||B|sin(theta) The Attempt at a Solution I know that to find the answer I need to find the cross product of C and D. I have done...

Homework Statement I need to prove the identity div (a x b) = b dot (curl a) - a dot (curl b) The Attempt at a Solution I've done the proof about 10 times now, and everytime I get the left hand of the identity equal to this: (all the d's are partial derivatives) d(a3b1)/dx -...

Hi. I have been looking at differential forms, and that inspired me to consider a partial derivative as a ratio between cross products. Please tell me if the following makes sense. Say we have cartesian coordinates (x,y) and polar coordinates (\rho, \phi). I want to calculate...

This question has a few parts. r = i + 2j + 3k s = 2i - 2j - 5k t = i - 3j - k Evaluate: a)(r.t)s - (s.t)r b)(r x s) x t. deduce that (r.t)s - (s.t)r = (r x s) x t can you prove this relative true for any three vectors a)(r.t)s - (s.t)r (r.t)s well I don't know...

Is there any physical meaning to the cross product of two magnetic fields e.g. two fields generated in two different current loops?

Hello! I'm trying to understand how this formula: 1/2 magnitude of v × w (area of triangle) yields the same value no matter which 2 adjacent sides are chosen. How would you prove mathematically that this is the case?

In the equation v^2 = u^2 + 2aS , What kind of products are v^2 , u^2 , and aS ? Cross product or dot product?

Homework Statement Hello. I am relatively new to this subject please forgive my incompetence. Please correct me if I have misunderstandings. I understand that the cross product of two vectors (say A and B) in R3 is a vector that is orthogonal to both A and B. But how A x B be...

How do I take the cross product of Two 3x3 Matrices. For example what is cross product of: [-1 0 0] [0 1 0] [0 0 1] x [0 -1 0] [1 0 0] [0 0 1] thanks, Della

Homework Statement If you make the change of variables: x= ucos(\theta)-vsin(\theta) y= usin(\theta)+vcos(\theta) where the angle 0 \leq \theta <\pi/2 is chosen in order to eliminate the cross product term in: 4x2+8xy+6y2=30 What is the angle you would use? Homework Equations...

I'm very confused by this cross product thing...help!

1. Find the sine of the angle between and . 2. AxB = ABsin(alpha) 3. I know T = (3,1,0) and U = (2,4,0), and I calculated that V = TxU = (0,0,10), so the magnitude of V is 10. Then I did 10 = sqrt(10)*sqrt(20)*sin(alpha) Divided 10 by sqrt(10)*sqrt(20), took the inverse...

Cross product and the angles of a prallelogram's diameters Homework Statement With the help of cross product prove: if the angle between the diameters of a parallelogram is 90 degrees, then the parallelogram is a rectangular. Homework Equations A\timesB=(surface of the parallelogram)...

Homework Statement Let \vec{A}, \vec{B}, and \vec{C} be three vectors which are all not in the same plane. Show that \vec{A}{\cdot}(\vec{B}{\times}\vec{C})=\vec{B}{\cdot}(\vec{C}{\times}\vec{A})=\vec{C}{\cdot}(\vec{A}{\times}\vec{B}) Homework Equations Don't know :( The Attempt at...

I'm studying vector calculus, and have a question about the curl and its relation to a cross product of the del operator and a vector. When doing a standard cross product as the formula I have i(det1) - j(det2) + k(det3), where det1, 2 3 are the appropriate 2x2 determinants. However for the...

Could anyone tell me how to use the cross product to prove the sine rule

Why is the cross product defined this way (blah)i + (blah)j + (blah)k = u and used this way (blah)i [SIZE="3"]- (blah)j + (blah)k = u

I am having problems with this questions my physics teacher gave us. Since he doesn't explain things well, I do not know how he got an answer of 66. Can you please explain to me how to get this answer? THX. For the following three vectors, what is C*(A X 3B) (the "*" is suppose to be a dot)...

Describe the surface defined by the equation: (a) \vec{r}\cdot \hat{x}= 2, where \vec{r}=x\hat{x}+y\hat{y}+z\hat{z}; (b) \left \| \vec{r} \times \hat{x}\right \|=2 For the first one, I know that is interpreted as the projection of the r vector onto the x-axis is equal to two...

I was reading div grad curl and all that. Couldn't get something basic - cross product. U = iux + k df/dx (ux) V = jvy + k df/dy (vy) UxV = [ -i df/dx - j df/dy + k] uxvy can't figure out how to get U x V.

Homework Statement show that three points P,Q, and R are collinear if and only if vec PQ x vec PR = 0 Homework Equations The Attempt at a Solution i understand this (i can see this visually in my head) but i just don't know how to say it mathematically...i know that the angle...

For the two matrices A and B, (AB)i,j = ri [SIZE="4"]. dj ---- [SIZE="4"]. refers to dot product ---- ri is the ith row in A and dj is the jth column in B. Let us say that A and B are n x n system of column vectors. Then a row vector ri of A would correlate to a component vector of the sum of...

Homework Statement I simply forget how to cross product if you have: 3(A x B) Would it be 3A x B or A x B then take that result and multiply by 3?

Homework Statement You are looking down at a map. A vector u with ||u|| = 5 points north and a vector v with ||v|| = 10 points northeast. The cross product u×v points: A) south B) northwest C) up D) down Please enter the letter of the correct answer: The magnitude ||u×v|| = Homework Equations...

Hi, I'm very ashamed about getting a fully understanding of these vector product concepts. But i did a lot of search and get an idea about them. I read Feynman's lecture on physics and almost completely understand the mathematics behind their properties. In that book feynman proves that cross...

in the end of this http://i29.tinypic.com/5fktjo.jpg i got to the point where it asks what B is i think that because i X J = K then b should be in the direction of J but they have a different conclusion why??

i need to figure out the size and direction of the force on each segment http://i32.tinypic.com/1075g8g.gif in cd segment there is a 45 degree between the electricity and magnetic field so i need to have size 40*sin 45 why they write only 40 ??

http://i26.tinypic.com/27zdaiv.gif i know that i should do right hand rule to know the result but here the vectors are weird ??

http://i25.tinypic.com/34s2flh.gif they say that the magnetic field is paralel to the plane so cross product is 0 (sin 0 =0) where is the mistake in my logic

I just saw an instructor of EEE (YouTube) talk about Cross Product. He gave an example of when cross products are used. Here's what he said. He drew a vertical arrow and said this is current. Then he drew a circular arrow around the vertical line and said it is B. Then he drew a smaller...

Homework Statement Which of the following functions is not an inner product in \mathbb{R}^2, where u=(a, b), v=(c,d)? a. <u,v> = ac+bd b. <u,v> = 4ad+7bc c. <u,v> = ac/2+bd/3 d. <u,v> = 2ac+9bd Homework Equations None The Attempt at a Solution u x v = det(u v) = ad-bc I...

Hello guys, this must be a very trivial question, but I just don't see it. Ihave a "vectorized quaternion" (quaternion with a null scalar part) : u(t) = (0, u_x(t), u_y(t), u_z(t) ) We also have it's time derivative : \frac{d}{dt}u(t) = \dot{u}(t) = (0, \dot{u}_x(t), \dot{u}_y(t)...

Regarding the equation for curl: Nable E literally means the sum of the differences of certain rates of change with respect to certain coordinates i hat, j hat, k hat. Since Nabla Cross E also is interpreted as the volume of a paralleliped in 3D space... 1. when the volume is zero, does...

I've been reading a bit on vector calculus trying to get a solid grasp on it. I am wondering how it can be stated that given C=AXB (all vectors), that C is equivalent to the area of the rectangle formed by A and B in the A,B plane. How can you equate length and area. One is in say...mm and the...

Hello, this is pretty straightforward. I need to take cross product: kq(x/(x^2+y^2),y/(x^2+y^2)) x (d/dx,d/dy) since kq is a scalar can i just leave it outside the calculation until the very end and for now just calculate the cross product of the two and once i get a definite answer...

simple cross product question. Please help! Homework Statement Hey guys, I know how cross product works, i-->j-->k : positive numbers i<--j<--k : negative number Homework Equations The question that I have is, say I have -k x -i, I am supposed to get a positive j since the two...

Homework Statement If theta is the angle between two non-zero vectors A and B, then which of the following angles theta results in A dot B = |A x B|? Homework Equations A dot B = ABcos(theta) A x B = ABsin(theta) The Attempt at a Solution There were two choices in the multiple choice answers...

qvxb maybe not?? Homework Statement consider the example of a positive charge moving in the -z direction with speed with the local magnetic field of magnitude in the +z direction. Find , the magnitude of the magnetic force acting on the particle. Express your answer in terms of , , , and...

Say A,B and C are points on a plane. By taking the magnitude of the cross product of AB and BC gives you the area of the parallelogram. The direction of the answer will be perpendicular to both AB and BC, but what I don't understand is why we are getting the area? Is the area pointing in that...

TO be very honest i only know the definition of cross product of two vectors but i would like to prove.....in fact all textbooks just define the cross product how can i derive it starting from the first principles i,e, i must start with something and finally arrive at the expression given...

Let vectors vec{A}=(0,0,-3), vec{B}=(2,0,0), and vec{C}=(0,1,1). Calculate the following, expressing your answers as ordered triples (three comma-separated numbers). Calculate A x C. For this part, remember that only perpendicular components contribute to the cross-product. The vector C has...

Hey all, I really need some clarification here. I've seen problems dealing with the Angular Momentum of a particle, working in spherical coordinates. Wolfram says that there is no simple way to perform this and do the determinant, and you will find many people and other websites claiming...

Homework Statement p X q - r Homework Equations The Attempt at a Solution In order to evaluate this expression, I must first evaluate q - r and then perform the cross product, right? I'm trying to think about X as an operator, like del, that operates on the things that appear...

If there are three nonzero vectors.. Do you think cosine effects this example:show the three vectors must lie in the same plane? ------------- * -->dot product X -->cross product -------------- A*(BXC)=0 so we can change as.. |A||B||C|sin\alphacos\beta=0 then... we can meet...

To me the idea of a force that pushes an object only in a "sideways" direction - that is, always at a right angle to the direction of motion - has always seemed strange. I can see how it comes about in the case of the Coriolis force because there the frame of reference is rotating and the force...