What is Cross product: Definition and 469 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



{\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



{\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. A

    Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

    I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
  2. W

    I The Vector Laplacian: Understanding the Third Term

    Suppose ##A = A_i\mathbf{\hat{e}}_i## and ##B = B_i\mathbf{\hat{e}}_i## are vectors in ##\mathbb{R}^3##. Then \begin{align} \Delta\left(A\times B\right) &= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\ &= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta...
  3. T

    I Why is the Cross Product Used in Mathematics? Understanding its Role and History

    So I do know that there does exist a generalization of the cross product (the exterior product), but this question does not concern that (and I would prefer it not ) I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only...
  4. C

    Deriving algebraic definition of cross product

    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!
  5. C

    Difference between scalar and cross product

    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...
  6. C

    Explaining the Cross Product for Two Vectors

    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!
  7. J

    Understanding the Dot Product and Cross Product in Vector Calculations

    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.
  8. tbn032

    B Confusion about the angle between two vectors in a cross product

    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...
  9. T

    A Reason out the cross product (for the moment): a skew symmetric form

    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...
  10. Garlic

    I Cyclic rotation of the cross product involving derivation

    Dear PF, so we know that cross product of two vectors can be permutated like this: ## \vec{ \alpha } \times \vec{ \beta }=-\vec{ \alpha} \times \vec{ \beta} ## But in a specific case, like ## \vec{p} \times \vec{A} = \frac{ \hbar }{ i } \vec{ \nabla } \times \vec{A} ## the cyclic permutation of...
  11. kuruman

    B A "no hands" rule for the cross product (requires literacy)

    Although it is considered unwise to judge a book by its cover, a book's cover is still useful for finding the direction of the cross product ##\mathbf{A}\times \mathbf{B}## between two given vectors. Being able to read is all that is needed. Here is a detailed procedure. Step 1. Move one...
  12. U

    Vectors in yz and xz plane dot product, cross product, and angle

    I tried to find the components of the vectors. ##a_y =2.60 sin 63.0 = 2.32## and assuming the z axis would behave the same as an x-axis ##a_z =2.60 cos 63.0 = 1.18## ##b_z =1.30 sin 51.0 = 1.01## making the same assumption ##b_x =1.3 cos 51.0 = 0.82## I now think I should have switched these...
  13. A

    I Time derivative of the angular momentum as a cross product

    I am trying to find the equations of motion of the angular momentum ##\boldsymbol L## for a system consisting of a particle of mass ##m## and magnetic moment ##\boldsymbol{\mu} \equiv \gamma \boldsymbol{L}## in a magnetic field ##\boldsymbol B##. The Hamiltonian of the system is therefore...
  14. D

    I Question about the vector cross product in spherical or cylindrical coordinates

    Hi If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b. Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
  15. A

    I Cross Product in E_u: Explaining Gourgoulhon's Text

    I study from Gourgoulhon's text 'special relativity in general frames', I have some difficulty to understanding Chapter 3 Page 84. I already learn that there exist a orthogonal projection mapping ##\bot_{u}:E \rightarrow E_u(P)## from the vector space ##E \cong R^4## to the subspace ##E_u(P)##...
  16. E

    B What is the cross product of two vectors?

    I understand that dot product gives us a number and cross product gives a vector. Why is this vector orthogonal to the others two, and why it has magnitude |a|*|b|*sinΘ? How to use cross product? What does it give to us?
  17. greg_rack

    Vector cross product anti-commutative property

    That may sound really silly, and that may be due to my lack of understanding of the operations itself, but: if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...
  18. greg_rack

    Calculating vector cross product through unit vectors

    Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates: $$ \vec{U}\times \vec{B}= (U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times (B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$ From this point on, I...
  19. Filip Larsen

    I Rotational invariance of cross product matrix operator

    Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...
  20. G

    I How to approach a cross product question

    I am beginning this new general physics course and I have encountered a question involved with what I assume to be cross products, a topic that I have very little experience with. I am not looking for a direct answer to the problem but advice on what steps should be taken in order to learn how...
  21. K

    Calculating crossproduct integral, Parametrization

    i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right? The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...
  22. G

    How to get the magnetic moment for this loop?

    About this figure, the current in the opposite wires are parallel (and not anti-parallel). So, for instance for the first option the torque is zero; but I wanted to know what is the magnetic moment of this loop. Since I rely only on formula I've have no idea how to compute for this one.
  23. N

    Why is the Lie Bracket the same as the Cross Product for a 2 Sphere in R3?

    Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y) [X,Y] = JYX - JXY where the J's are the Jacobean matrices. I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...
  24. J

    Vector Cross Product With Its Curl

    Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...
  25. P

    Stuck on a few Vector homework problems

    I'm stuck on a few Vector homework problems. I don't quite understand how to write vectors A+B and A-B for questions 1b and 2b. I tried starting with calculating the magnitude for vector A+B on question 1b and then followed by finding theta, but I'm not sure if that's what I'm supposed to do...
  26. Avatrin

    I Understanding the cross product and rotations

    Hi I have used cross products thousands of time without really knowing what it actually does; I know how to compute it, but I don't feel like I understand it. Also, when it shows up in physics/kinematics contexts, it's only because the magnitudes of the vectors involved have to be multiplied...
  27. S

    I Can the cross product concept be completely replaced by the exterior product?

    Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...
  28. hnnhcmmngs

    But, as I said, you don't actually need the coordinates at all.

    Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...
  29. D

    I Relation Between Cross Product and Infinitesimal Rotations

    Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...
  30. S

    Calculate Torque on a Particle with Given Force and Position | Homework Solution

    Homework Statement Force F⃗ =−11j^N is exerted on a particle at r⃗ = (8i^+5j^)m. What is the torque on the particle about the origin? Express your answer using two significant figures. Enter coordinates numerically separated by commas.Homework EquationsThe Attempt at a Solution F: 0i, -11j...
  31. Specter

    Why does the cross product of two direction vectors....

    Homework Statement [/B] Hopefully this is in the correct section I looked around for others but this seemed like the right one. Find the scalar, vector, and parametric equations of the plane that passes through the points P(1,0,4), Q(3,1,-6), and R(-2,3,5). Homework EquationsThe Attempt at a...
  32. Hawkingo

    B How to specify the direction of an area vector?

    We all know that the area of a triangle having consecutive sides as ##\vec { a }## and ##\vec { b }## has the area ##\frac { 1 } { 2 } | \vec { a } \times \vec { b } |## but what is the direction of that area vector? I mean if we consider ##\vec { a } \times \vec { b }## that will be one...
  33. J

    Angular momentum relative to the origin

    Homework Statement A 2.4 kg particle-like object moves in a plane with velocity components vx = 25 m/s and vy = 80 m/s as it passes through the point with (x, y)coordinates of (3.0, −4.0) m. (Express your answers in vector form.) (a) What is its angular momentum relative to the origin at this...
  34. maxknrd

    I More elegant way to solve divergence of arbitrary dotproduct

    This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
  35. JTC

    I Why does the cross product operation adhere to the right-hand rule?

    (First, I am aware there is no cross product in higher dimensions. I am aware of differential forms. I am aware of the problems with the cross product. I know enough to know what I don't know. but set all that aside.) With that, could someone help me, by properly phrasing the cross product...
  36. Sandro Romualdez

    Vectors: Proving the value of a•(b x c)

    Homework Statement Given that vector a = (1, 2, -5), b = (-12, 41, 75) and c = a + 2b, explain why (without doing any calculations whatsoever) the value of a•(b x c) = 0 Homework Equations No specific equations, as the question asks for the value without making any calculations. This problem...
  37. Dimani4

    Define the polarization in TE/TM modes

    Hello people, I'm a little bit confused about how to define the polarization direction for TM/TE mode. Take a look at the TE mode picture I found in some place. Picture1 The Cartesian system of coordinate (XYZ) here is chosen by the right hand rule. Picture2 But how we chose the direction...
  38. Philosophaie

    I Is this 4-Vector cross product calculation correct?

    Could someone tell me if this 4-Vector cross product is correct: i j k t dx dy dz 1/c*dt Ex Ey Ez Et =[(dy(Ez)-dz(Ey))-(dy(Et)-1/c*dt(Ey))+(dz(Et)-1/c*dt(Ez))]*i -[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*j...
  39. Vectronix

    B Derivative with the double cross product

    Is there a spatial derivative that uses the del operator and the double cross product? If so, is it used in physics?
  40. Wrichik Basu

    B What is the real life utility of the dot product and cross product

    Today, my teacher asked us what is the real life utility of the dot product and cross product of vectors. Many of us said that one gives a scalar product, and one gives a vector product. But he said that, that was not the real life utility of the dot and cross product. He asked us, "Students...
  41. ltkach2015

    A LeviCivita in Orthogonal Curvilinear Coordinate System: "Cross Product Matrix

    If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same? "Levi Civita 3x3x3 is said to be independent of any coordinate system or metric...
  42. JTC

    The Cross Product and Angular Momentum

    Hello I need help to explain the affect of the cross product without the its current symbolism, but for angular momentum. I can explain angular momentum in terms of the cross product of 3D space formulated like this: |r| |v| * sin(angler.v) e-perp to r and v Eq.1 (I can explain this to...
  43. C

    Calculating the Cross Product Vector: Is It Possible to Find VxW?

    Okay so, I am wondering if it is possible to find the actual cross product (not the magnitude of the cross product) from this information 1. magnitude of both vectors 2.angle between vectors 3.plane the vectors lie in Is there any way to calculate that cross product vector? Thank you very much...
  44. 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
  45. F

    I Can anybody give me a list of all cross products in physics.

    Can anybody give me a list of all cross products in physics. I have the following in my list: Torque $$\vec{\tau}=\vec{F}\times\vec{r}$$ Angular momentum $$\vec{L}=\vec{r}\times\vec{p}$$ Velocity $$\vec{v}=\vec{\omega}\times\vec{r}$$ Biot-Savart law $$\vec{dB}=\dfrac{\vec{i...
  46. Jovy

    What is the sine of the angle between two given vectors?

    Hello, I apologize in advance for the way this post looks. I am new to this forum and I've never used LaTeX Primer. I noticed that someone has prevoiusly asked the same question, but I still do not understand how to get to the answer. Also, I tried posting an image but I could not; and this...
  47. bwest121

    I Question about Cross Product and Derivative

    Hi everyone, Given a vector-valued function ##\vec{A}##, how do I show that: $$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$ In other words, are the cross product and derivative commutative w/ each other? I...
  48. Surya97

    Is there a true upward force on a rotating object?

    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...