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.)
I have a nice table that shows the dot product between unit vectos (see annex). I'd like know how is the cross product between unit vectos of all basis. Do you have a table with such information?
Dear all,
I've encountered some problems while looking through the book called "Operator Algebras" by Bruce Blackadar.
At the very beginning there is a definition of pre-inner product on the complex vector space: briefly, it's the same as the inner product, but the necessity of x=0 when [x,x]=0...
Homework Statement
Find the inner product of f(x) = σ(x-x0) and g(x) = cos(x)
Homework Equations
∫f(x)*g(x)dx
Limits of integration are -∞ to ∞
The Attempt at a Solution
First of all, what is the complex conjugate of σ(x-x0)? Is it just σ(x-x0)?
And I'm not sure how to...
i realize this is a linear algebra question, but the bra-ket notation is still a little confusing to me so i posted it in this section.
|e>=(1+i,1,i) (n-tuple representation, where i's are the imaginaries)
so the norm of this would then be the following...
Homework Statement
If the 2.0 x 10-5 mol of Cu(IO3)2 can dissolve in 2 L of NaIO3, find the molar concentration of the NaIO3 solution. Ksp = 1.4 x 10-7 for Cu(IO3)2.Homework Equations
The Attempt at a Solution
Let y = [IO3-(aq)] present in the solution from NaIO3 Cu(IO3)2(s) ↔ Cu2+(aq) +...
Show that
\[\prod_{k=2}^{\infty} \left(\frac{2k+1}{2k-1}\right)^{k} \left(1-\frac{1}{k^2}\right)^{k^2}=\frac{\sqrt{2}}{6}\pi \]
This problem can be solved using only elementary methods. :D
If $a,b,c,d$ are distinct real no. such that
$a=\sqrt{4+\sqrt{5+a}}\;,b=\sqrt{4-\sqrt{5+b}}\;,c=\sqrt{4+\sqrt{5-c}}\;,d=\sqrt{4-\sqrt{5-d}}$. Then $abcd=$
Hey MHB, I am back, fully recover from food poisoning and first off, I want to take this opportunity to wish everyone and their family a very happy and Merry Christmas, much luck, good health and all good things of life. I hope you guys are able to spend it with loved ones!(Inlove)
I want to...
Hi All,
Have the following given and would like some help with the 3 questions below. Any help will be greatly appreciated :)
City 1
TOTAL POPULATION CITY 1 100,000
Avg Age of Population 27.8
Median Age (Male) 20.5
Median Age (Female) 27.8
% Male Population 46.43%
% Female Population 53.57%...
I'm so confused about finding an angle, theta in this illustration.
With having three coordinate information, how can I calculate the theta using dot product?
I would easily find the angle by using trigonometric formula if I ignore the z-axis.
But I want to solve this problem with...
I was thinking, if exist a product (cross) between vectors defined as:
\vec{a}\times\vec{b}=a\;b\;sin(\theta)\;\hat{c}
and a product (dot) such that:
\vec{a}\cdot\vec{b}=a\;b\;cos(\theta)
Why not define more 2 products that result:
\\a\;b\;sin(\theta) \\a\;b\;cos(\theta)\;\hat{d}
So, for...
True or False, if AxB = AxC then either A=0 or B=C.
A, B, and C are vectors and I thought this statement would be true. However the answer key says it is not. Why?
Hi,
Short question: If you take the inner product of two arbitrary wave functions, and then the gradient of that, the result should be zero, right? (Since the product is just a complex number.) Am I missing something?
∇∫dΩψ_{1}*ψ_{2} = 0
I am studying relativity by myself. There is one problem in the book which says that the 4-dot product of the Minkowski force and proper velocity is zero. But again it say that qE.u = change in energy over time. Is there a contradiction? If not, Am I missing something important.
here q is...
Homework Statement
Given ##S = \{1, x, x^2\}##, find the coordinates of ##x^2 + x + 1## with respect to the orthogonal set of S.Homework Equations
Inner product on polynomial space:
##<f,g> = \int_{0}^{1} fg \textrm{ } dx##
The Attempt at a Solution
I used Gram-Schmidt to make ##S## orthogonal...
Hi,
I know these questions must sound ridiculous and I apologize, I'm a newbie. My textbook says that the inner product of the momentum four-vector is
P\bulletP=P\bulletP - E^{2}/c^{2}=-m^{2}*c^{2}
So my silly questions are: 1) where did the - E^{2}/c^{2} term come from? 2) I know I'm being...
Homework Statement
a_n is a sequence of positive numbers. Prove that \prod_{n=1}^{\infty} (1+a_n) converges if and only if \sum_{n=1}^{\infty} a_n converges.
Homework Equations
The Attempt at a Solution
I first tried writing out a partial product: \prod_{n=1}^{N} (1+a_n) =...
Can I say that \eta^{ij}\eta_{km}=\delta^{i}_{k}\delta^{j}_{m}?
Kind of in the same way that they yield one delta in the case where one of their indices is summed over?
Thanks
Hello
I have a set of sets of real numbers greater than 1. Each set can have a different quantity of numbers.
Set A1 {a11, a12,...a1m1}
Set A2 {a21, a22, ..., a2m2}
...
Set AN {aN1, aN2, ..., aNmN}
If I want the sum of all possible products that have one element from each set, that's...
The Vector A points 17° counterclockwise from the positive x axis. Vector B lues in the first cuadrant of the xy plane. The magnitudes of the cross product and the dot product are the same:
i.e, |AXB|= |A(times)B|
What Angle does B make with the positive x axis?
2. Is ti a scalar...
Consider the function APL=\frac{\sqrt[4]{L}}{L}, where L is the number of workers. The company has just hired 8 workers. What is the marginal product of the labor?I know that if I had the total product I could differentiate it and get the marginal product, but it's the average product that is...
Dear Forum :
I hung up with a integration
http://ppt.cc/mIpV
Can it be deduced to a simpler form?
The distribution of σ(E) is http://ppt.cc/-5Z5
The estimation width of x is 10MeV , height is 200mb.
The distribution of dE/dx is http://ppt.cc/vcVU
Is there a way to do some simple...
Homework Statement
Assume A and B are normal linear operators [A,A^{t}]=0 (where A^t is the adjoint)
show that det AB = detAdetB
Homework Equations
The Attempt at a Solution
Well I know that since the operators commute with their adjoint the eigenbases form orthonormal sets...
I was working on a pde, and I needed to compute a Jacobian for it.
Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a matrix, and b is...
I was working on PDE for a project and needed to compute a Jacobian for it.
Suppose we have a function consisting of a series of matrices multiplied by a vector:
f(X) = A * B * b
--where X is a vector containing elements that are contained within A, b, and/or b,
--A is a matrix, B is a...
Hi,
So I still not sure how to apply like rhr rule in this setup in problem like the one in the following so I tried to do rhr in order to get the direction but it didn't work out. this is an example from halliday and resnick book.
Figure 32-24 shows a wire segment,placed in a uniform...
Just for fun, eh...? (Heidy)For z \in \mathbb{R}, and m \in 2\mathbb{N}+1, show that:\frac{\tan mz}{\tan z}=\prod_{j=1}^{ \lfloor m/2 \rfloor } \tan\left(\frac{j\pi}{m}+z\right) \tan\left(\frac{j\pi}{m}-z\right)
Homework Statement
Given 3 measure spaces (X,A,\mu), (Y,B,\zeta), (Z,C,\gamma), show that the product of the three sigma algebras A, B, and C is associative, meaning that:
AxBxC=(AxB)xC=Ax(BxC)
Homework Equations
We can make use of the fact that XxYxZ=(XxY)xZ=Xx(YxZ)
The Attempt at a...
I was wondering about the proper way to say, \langleA|B\rangle .
I have recently heard, "The inner product of A with B." But I'm not sure if this is correct. Does anyone know the proper order in which to place A and B in the sentence?
As a simple example: Suppose you're speaking with...
Suppose I was asked if G \cong H \times G/H . At first I considered a familiar group, G = S_3 with its subgroup H = A_3 . I know that the quotient group is the cosets of H, but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't...
Let f:\mathbb{R} \to \mathbb{R} and g:\mathbb{R} \to \mathbb{R} be discontinuous at a point c . Give an example of a function h(x)=f(x)g(x) such that h is continuous at c.
f(x) =
\begin{cases}
0 & \text{if } x \in \mathbb{Q} \\
1 & \text{if } x \in \mathbb{R}-\mathbb{Q}...
Homework Statement
For what values of k is (scalar product of vectors a and b) = a_{1}b_{1}-a_{1}b_{2}-a_{2}b_{1}+ka_{2}b_{2} a valid scalar product?
The vectors a and b are defined as:
a = a_{1}e_{1} + a_{2}e_{2}
b = b_{1}e_{1} + b_{2}e_{2}
where e_{1} and e_{2} are unit vectors...
I am currently going through the book Introduction Of Electrodynamics by Griffiths. I have come across vector triple product which is stated as follows in the book:
$$\textbf{A} \times (\textbf{B} \times \textbf{C})=\textbf{B}(\textbf{A}\cdot \textbf{C})-\textbf{C}(\textbf{A}\cdot...
Let's say we have operator X that is Hermitian and we have operator P that is Hermitian. Is the following true:
[X,P]=ihbar
This is the commutator of X and P.
This particular result is known as the canonical commutation relation.
Expanding:
[X,P]=XP-PX=ihbar
This result indicates that...
Hi, I was looking for a proof or explanation of this. From Schroeder's Thermal Physics, pg 56, explaining interacting systems in equilibrium.
The example in the text is two 3-harmonic oscillators with a total of 6 units of energy. So one macrostate is where each has 3 units of energy. The...
Hello
I'm not sure if this belongs in this forum or the homework forum, but I have a quick question about selectivity and yield in reactions where one product is made, and that product returns to one of the reactants to create another separate reaction.
for example:
C_{6}H_{12} + H_{2}O...
We work on optical simulation where we use not ideal beam expander.
Not ideal means that for beam expander designed for single mode (M^2=1), the output beam has M^2 >1 (M^2 = M squared)
In our system we want to use beam expander with multimode laser beam.
The beam expander is not ideal (for...
Hi,
can somebody help me with the problem:
Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows...
Hello, I have a quick question about integrals of dot products. We are learning about magnetic flux as the integral of b dot da. However, what circumstances must be present where we can simplify this integral into (b*a) and ignore the integral?
What is the difference between a dot product and an inner product. The internet says that they are generalizations of each other. What does that even mean? Thanks for any help.
Homework Statement
Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...
How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following:
eb.\partialcea=-\Gammaa bc
How do I get just an expression for \partialcea?
I would like to know why $M_n$ $\not\cong$ $O_n$ x $T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations in $\mathbb R^n$.
**My attempt:** Can I show that one side is abelian, while the other group is...
Homework Statement
Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the...