What is Vector product: Definition and 49 Discussions

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.)

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  1. S

    B Geometrical meaning of magnitude of vector product

    My notes says that the geometrical meaning of $$|\vec v \times \vec w | $$ is the perpendicular distance from point ##V## to line passing through ##O## and ##W## (all vectors are position vectors) $$|\vec v \times \vec w | = |\vec v| |\vec w| \sin \theta$$ From the picture, the perpendicular...
  2. M

    A Tensor and vector product for Quantum

    Hello, I am calculating the krauss operators to find the new density matrix after the interaction between environment and the qubit. My question is: Is there an operational order between matrix multiplication and tensor product? Because apparently author is first applying I on |0> and X on |0>...
  3. L

    A Commutator vector product

    Is there some easy way to see that [\vec{p}^2, \vec{p} \times \vec{L}] is equal zero? I use component method and got that.
  4. Andrea Vironda

    I Can the Differential Pass in the Beginning Part of a Change in Vector Product?

    Hi, In a demostration i found a change in order i can't understand. how can the differential pass in the beginning part? the only thing I'm sure is that "v" and "dr" are parallel
  5. davidge

    I Why this form of the vector product?

    Hi. I was investigating through this week why there are the differential forms, why are they anti-symmetric, why do we have the Jacobian when expressing the volume in a different coordinate system. This was just fantastic! I found all the connections between these topics. And I found that all of...
  6. andylatham82

    I How to find angle between vectors from dot and cross product

    Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy
  7. andylatham82

    I Angle between vectors via scalar product vs vector product

    Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...
  8. N

    I A new vector-product for geometric algebra?

    I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)? I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say...
  9. Muthumanimaran

    Vector Cross Product Homework: Expand $\vec{v}\times({\nabla}{\times}\vec{A})$

    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...
  10. prashant singh

    I Cross product or vector product.

    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...
  11. prashant singh

    I Scalar product and vector product

    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...
  12. ognik

    MHB Triple vector product with del

    I know the bac-cab rule, but add $\nabla$ and it's not so clear .. applying it to $\nabla \times \left( A \times B \right) = A\left(\nabla \cdot B\right) - B\left(\nabla \cdot A\right) ...$, not quite Please walk me through why the other 2 terms emerge ?
  13. T

    What Is the Correct Angle Between Two Vectors If Their Cross Product Is Known?

    Homework Statement Two vectors A and B have magnitude A = 3.00 and B = 3.00. Their vector product is A x B= -5.00k + 2.00i. What is the angle between A and B? Homework Equations Magnitude of vector product = magnitude of A * magnitude of B * sin of the smaller angle between A and B...
  14. PcumP_Ravenclaw

    Solution to vector equation involving vector product

    I want to find the solution of vector X. I am using text from Alan F. Beardon Algebra and Geometry as attached. I don't know how the solution is derived for the following equation. ## x + (x × a) = b ## The second solution when ## a \times b \neq 0 ## then X cannot be b. Is it possible to...
  15. S

    Prove that curl of a vector is a vector

    Homework Statement Proove it Iam supposed to change coordinate system, and proove that the result depends on coordinate system. The Attempt at a Solution My attempt was to start from definition of cross product using levicivita. I already prooved that divergence of a vector is a scalar. But...
  16. N

    What is the variance of the product of a complex Gaussian matrix and vector?

    Hi, Assuming that A is a n x m random matrix and each of its entries are complex Gaussian with zero mean and unit-variance. Also, assume that b is a n x1 random vector and its entries are complex Gaussian with zero mean and variance=s. Then, what would be the variance of their product Ab? Any...
  17. Bassa

    Vector product and vector product angles

    Hello! I have a problem in my calculus based physics class regarding vectors. The problem says: Vectors A and B have a scalar product -6.00 and their vector product has magnitude 9.00 what is the angle between these two vectors? Here is how I approached it: -6=|A||B|cos (theta) 9=|A||B|sin...
  18. Telemachus

    Identity, vector product and gradient

    Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...
  19. J

    The Integration-By-Parts Formula for Vector Products

    We know that: \frac{d}{dx}(\vec{f} \cdot \vec{g}) = \frac{d\vec{f}}{dx} \cdot \vec{g} + \vec{f} \cdot \frac{d\vec{g}}{dt} and: \frac{d}{dx}(\vec{f} \times \vec{g}) = \frac{d\vec{f}}{dx} \times \vec{g} + \vec{f} \times \frac{d\vec{g}}{dt} But, exist some formula (some expansion) for: \int \vec{f}...
  20. skate_nerd

    MHB Tensor notation for vector product proofs

    I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about. For example, I have a problem where...
  21. R

    Finding the Eigen Value of A.σ Vector Product

    Can anyone tell me what is eigen value of product of a vector with pauli matrices i.e A.σ where A is an arbitrary vector ?
  22. S

    Multiplying a Vector Product by Another Vector

    Homework Statement The question asks to calculate (AxB)·C, where A's magnitude is 5.00, B's magnitude is 4.00, and they are both in the xy-plane. B is 37° counter clockwise from A. C has a magnitude of 6.00 and is in the +z-direction. Homework Equations (A×B) = ABsinθ = D; D·C =...
  23. Battlemage!

    Why is the Vector Product Operation Considered Multiplication?

    The operation is called "the cross product." I am wondering if there are any specific characteristics that has led mathematicians to consider it a type of multiplication. After taking abstract algebra, the only conclusion I can come away with is that it more closely resembles traditional...
  24. G

    Vector product question in cylindrical coordinates

    I am trying to work the following problem; A rigid body is rotating about a fixed axis with a constant angular velocity ω. Take ω to lie entirely on th z-axis. Express r in cylindrical coordinates, and calculate; a) v=ω × r b)∇ × v The answer to (a) is v=ψωρ and (b) is ∇ × v = 2ω...
  25. D

    Vector 'B' When Vector Product with 'A' is Zero

    Homework Statement Vector 'A' is along postive z-axis and its vector product with another vector 'B' is zero , then vector 'B' could be .. a) i + j b) 4i c) i + k d) -7k . Homework Equations The Attempt at a Solution
  26. D

    Calculating Vector Product of A and B

    Homework Statement Given A= i + j (caps) and B=i+k (caps)..what is the value of vector product of A and B?Homework Equations The Attempt at a Solution .... i tried but can't get this one
  27. D

    Divergence of a tensor vector product

    can anybody tell me the expansion for the divergence of tensor vector product \nabla.(\tilde{K}.\vec{b}) for the case of scalar and vector the expansion is given by \nabla.(a\vec{b})=a\nabla.\vec{b}+\vec{b}.\nabla a
  28. S

    Understand Vector Product with Levi Civita Symbols

    Hello I am a graduate student and it is my first year in physics. Our instructor showed vector product with Levi Civita symbol and I did not understand the part where the product of two Levi Civita symbols is expressed as Kronecker Deltas. Additionally I did not get the proof on the URL...
  29. N

    Matrix vector product and linear transformation proof

    Homework Statement Hello! Prove: A(\vec{a}+\vec{b}) = A\vec{a} + A\vec{b} Where A is a matrix and T (in the following section) is a transformation. Homework Equations T(\vec{a}) + T(\vec{b}) = T(\vec{a}+\vec{b}) T(\vec{a}) = A\vec{a} T(\vec{b}) = A\vec{b} The Attempt at a...
  30. Q

    Vector product VS dot product in matrix

    hi, i don't really understand what's the difference between vector product and dot product in matrix form. for example (1 2) X (1 2) (3 4) (3 4) = ? so when i take rows multiply by columns, to get a 2x2 matrix, i am doing vector product? so what then is dot producT? lastly...
  31. M

    Vector Product Question (Mathematical Methods in Physics Course)

    Hey guys, I can't solve the first two problems. Can you help me please ASAP? http://img694.imageshack.us/img694/5562/hw1e.jpg
  32. W

    How to genarelize the vector product in three dimensions to higher dimensions?

    it seems that the vector product between vectors in three dimensions is peculiar property of the three dimensional space
  33. jegues

    Vector product and angles

    Homework Statement Two vectors lie their tails at the same point. When the angle between them is increased by 20 degrees the magnitude of their vector product doubles. The original angle between them was about _____ ? Homework Equations \overline{a}\bullet\overline{b} = |\overline{a}|...
  34. C

    Physical meaning of Vector product

    Can someone please explain to me why it is that the direction of a cross product is perpendicular to the plane of the original vectors? What is the physical significance of this? I mean, if I take torque as an example, what the hell does it mean to say that the direction of the torque is...
  35. A

    Understanding the vector product

    Homework Statement I've recently encountered the cross-product while studying mathematics. I'm studying on my own so it has been quite difficult to get a proper answer, which is why I'm posting my question here. What I've difficulties understanding is why the vector product of two vectors...
  36. J

    Where Can I Find Proofs for Vector Calculus Identities?

    Wikipedia gives an extensive amount of vector identities: http://en.wikipedia.org/wiki/Vector_calculus_identities http://en.wikipedia.org/wiki/List_of_vector_identities Does anyone know of a link where most of these are proved. I'm particularly interested in the product rules but would...
  37. A

    What is the net torque on a wheel with given dimensions and direction?

    Homework Statement find the net torque on the wheel about the axle through O, taking a = 7.00cm and b = 20.00cm. Assume the positive direction is counterclockwise Homework Equations torque = force x distance torque = force x (radius x sin(phi)) The Attempt at a Solution 10.0 x .2m...
  38. I

    Calculating the vector product

    i have a question I'm trying to find the cross product of the vector but don't know which formula to use. the first one is i(...) + j(...) + k (...) and the other one is the same as this one but has a negative sign i(...) - j(...) + k (...) what is the difference and which formula...
  39. K

    Vector Product in C³: Explained & Standard Basis

    The vector product in C³ is a three dimensional Lie algebra. Taking the standard basis (e_1,e_2,e_3) of C³, the brackets can be defined by the relations: [e_1,e_2]=e_3 [e_1,e_3]=-e_2 [e_2,e_3]=e_1 That what my book says, but I don't get. But what does the author mean here with the...
  40. B

    Surface defined by vector product

    Homework Statement Identify the following surface. I'm not sure what the answer is, but I believe I am on the right track, appreciate if anyone can check this for me. {\bf r} \cdot {\bf u} = m |{\bf r}| , -1 \le {\bf m} \le 1 Homework Equations Nothing really. The Attempt at a...
  41. S

    I evaluating this vector product identity

    Homework Statement The problem is written as: Del X (A X B) = (B*DEL)A- (A*DEL)B +A(DEL*B) -B(DEL * A) where * = dot. I don't know how to evaluate this because if the author meant for the standard mathematical order of operations to apply it makes since they wouldn't have worried about...
  42. D

    Differential Equation with Vector Product

    If a particle with charge e and mass m is in an arbitrary magnetic field has motion described by: m\frac{d^2\vec{r}}{dt^2}=\frac{e}{c}\frac{d\vec{r}}{dt}\times\vec{H} prove that the speed v\equiv\left\vert\frac{d\vec{r}}{dt}\right\vert is constant. I don't understand how to do this...
  43. X

    Find Scalar 'a' for Perpendicular Vectors L and K | Vector Product Homework

    Homework Statement Consider the two vectors L= i +2j+3K K=4i+5j+6k Find scalar 'a' such that: L - aK is perpndicular to L. Homework Equations if two vectors are perpenicular dot product=0 The Attempt at a Solution (i+2j+3k).{(1-4a)i+(2-5a)j+(3-6a)k}=0 I get three values of a...
  44. S

    Vector Product Direction & Magnitude Calculation

    Homework Statement vector A has magnitude of 2.80 m, 60 degrees above x axis. vector B has magnitude of 1.90 m 60 degrees below x axis. Is the direction of the vector product positive z direction or negative z direction? What is the magnitude of vector B times vector A. I already got...
  45. O

    Vector product - operator's a vector?

    Greetings all! So, \nabla X \vec{F} is confusing me. I understand that it can be used to tell whether a force is conservative in that, if the curl is 0 then the work done all all paths are the same... that's fine. However, I was looking at it, for example, in the context of the...
  46. E

    What is the magnitude of the vector product?

    Homework Statement Find the vector product \vec{A} \times \vec{B} (expressed in unit vectors ) of the two vectors given. What is the magnitude of the vector product? \vec{A} = 5.00\hat{i} + 2.00\hat{j} \vec{B} = 3.00\hat{i} - 1.00\hat{j} Homework Equations \vec{A}\vec{B}\sin(\theta)...
  47. J

    How was the vector cross product derived?

    In my calc book the derivation of the vector cross product is not derrived but rather just given. I've read in another book that William Rowan Hamilton, after years of work, came up with the basic form we momorize today and symbolize with determinants. Does anybody know how this vector cross...
  48. B

    Confusing arken/weber problem (del in triple vector product)

    the following problem is in the arfken/weber textbook and was also on a practice exam for my mathematical methods course: Verify that \mathbf{A} \times (\nabla \times \mathbf{A}) = \frac{1}{2} \nabla(A^2) - (\mathbf{A} \cdot \nabla)\mathbf{A}. i used the BAC-CAB rule, but i don't get...
  49. D

    What is the 8D Cross Vector Product?

    8d cross vector product,URGENT i wonder what will be the 8-dimensional cross vector product or (N1,N2,N3,N4,N5,N6,N7,N8)x(M1,M2,M3,M4,M5,M6,M7,M8)=? i need the answer real bad,so please answer if you can. if you happen to have the mathlab program it might answer cause it does all the...
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