# What is Product: Definition and 1000 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.)

View More On Wikipedia.org
1. ### My emphasis has always been on simplicity in my projects

I made the "pulser pump" https://en.wikipedia.org/wiki/Pulser_pump (Specifically to pump water from a small stream to water my garden without any effort.) Following that, I emigrated from Ireland to Canada, and did some work on very low pressure airlift pumps. I currently use air at about 1...
2. ### Obtain an equation of the plane in the form ##px+qy+rz=d##

The solution is here; Now to my comments, From literature, the cross product of two vectors results into a vector in the same dimension. A pointer to me as i did not know the first step. With that in mind and using cross product, i have ##(1-1)i - (-1-1)j+(1+1)k =0i+2j +2k## as shown in ms...
3. ### Chemical Composition / Ingredients to create a product similar to RMR 86

Hello Everyone, Im interested in creating a similar product to RMR 86, please see the SDS below. I was hoping someone can figure out which powder ingredients I need and in what amounts to mix with water ? It contains "UltraPure Sodium Hypochlorite"...
4. ### A Why does the description of a composite system involve a tensor product?

Can anyone answer me that why the description of composite system involve tensor product ? Is there any way to realize this intuitively ?
5. ### Product Cross Section For Zn66 in Geant4

Dear experts, I want to calculate the cross section (Product Crosssection) of the Ga66 element produced by sending deuteron to the Zn 66 element, which is my target in Geant 4.Is there a Geant4 sample file where I can do this calculation, so in which sample file can I calculate it? Do I need to...
6. ### About semidirect product of Lie algebra

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...
7. ### POTW Product of Two Finite Cyclic Groups

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]##.
8. ### Why is Gravitational Force Proportional to Mass Product?

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........
9. ### I Infinite product representation of Bessel's function of the 2nd kind

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...
10. ### 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!
11. ### I Absolute value bars in dot product derivation

Dose someone please know why they have the absolute value bars in this derivation? many thanks!
12. ### I Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors

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
13. ### I Dot product of two vector operators in unusual coordinates

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...
14. ### I Understanding tensor product and direct sum

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...
15. ### I Calculating Mass Inertia Product - Examples 1 & 2

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
16. ### I Question about implication from scalar product

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...
17. ### Expectation of Product of three RVs

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}?
18. ### I Limit of the product of these two functions

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...
19. ### 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...
20. ### 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!
21. ### POTW Semisimple Tensor Product of Fields

Let ##L/k## be a field extension. Suppose ##F## is a finite separable extension of ##k##. Prove ##L\otimes_k F## is a semisimple algebra over ##k##.
22. ### 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.
23. ### B Tensor product of operators and ladder operators

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...
24. ### Why does the dot product in this solution equal zero?

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...
25. ### Every S-composite can be expressed as a product of S-primes

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...
26. ### POTW A Modified Basis in an Inner Product Space

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##.
27. ### 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...
28. ### I Inner product - positive or positive semidefinite?

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
29. ### I Inner and Outer Product of the Wavefunctions

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...
30. ### POTW Finding the Product of Real Roots: POTW Equation Solution

Find the product of real roots of the equation ##x^2+18x+30=2\sqrt{x^2+18x+45}##.
31. ### Factor the repunit ## R_{6}=111111 ## into a product of primes

Consider the repunit ## R_{6}=111111 ##. Then ## R_{6}=111111=1\cdot 10^{5}+1\cdot 10^{4}+1\cdot 10^{3}+1\cdot 10^{2}+1\cdot 10^{1}+1\cdot 10^{0} ##. Note that a positive integer ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ## where ## 0\leq a_{k}\leq 9 ## is divisible by ## 7, 11 ##, and...
32. ### Proving that ##T## is skew-symmetric, inner product is an integration.

##\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)...
33. ### MHB Minimum of product of 2 functions

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
34. ### Solve the quadratic equation that involves sum and product

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##...
35. ### Rotating object using product of two quaternions

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...
36. ### A Tensor product matrices order relation

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 am...
37. ### I How to visualize 2-form or exterior product?

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?
38. ### I Is tensor product the same as dyadic product of two vectors?

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...
39. ### Is result of vector inner product retained after matrix multiplication?

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...
40. ### Every integer n>1 is the product of a square-free integer?

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} ##...
41. ### I Wedge product of a 2-form with a 1-form

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...
42. ### I Sum of the dot product of complex vectors

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}##...
43. ### I A doubt regarding position representation of product of operators

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...
44. ### How do I calculate the work done by a force field using the dot product?

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?
45. ### I Confused about dot product of a and b = |a||b| if theta = 0

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?

47. ### Dot product: ##\vec{D} \cdot\vec{E}## in SI units

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?
48. ### Solve the quadratic equation involving sum and product

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...
49. ### Solve the given quadratic equation that involves sum and product

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##...
50. ### I Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

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...