Questions tagged [cross-product]
In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.
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What is the cross product
I know that the cross product of 2 points
$(1,2)$ and $(a,b,c)$ is $\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$
But what is the cross product of $(0,2)$ and $(1,3]$.
Do I need to take the different ...
4
votes
2
answers
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What is the operation between a bivector and a vector that outputs another vector?
I'll start with a physical example. Let's say we have the angular velocity (in 3D euclidean space with orthonormal basis spanning it), which has the bivector representation $$\Omega = \omega_x \mathbf ...
- 3,360
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Show that the sum of these four vectors is $0$.
Four vectors are erected perpendicularly to the four faces of a general tetrahedron. Each
vector is pointing outwards and has a length equal to the area of the face. Show that the sum of these four ...
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Cartesian product of two balls
Consider the following exercise:
Let $(a,b)\in\mathbb{R}^n\times\mathbb{R}^m$ be an arbitrary point and let $\varepsilon>0$. Show that there are positive real numbers $\varepsilon_1,\varepsilon_2$ ...
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This function has a fixed point, $F^4(x)=x$. Why?
Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
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Explanation of norm of cross product formula
I have a line between two points given as $(x_1, y_1)$ and $(x_2, y_2)$ in Python code:
...
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Can you characterize a vector field $v$ with uniform curl as $v(x) = a + b \times x$?
Let $v:\mathbb R^3 \to \mathbb R^3$ a vector field such that $\forall x\in\mathbb R^3, (\nabla \times v) (x) = \rho~$ where $\rho$ is a constant. Example here.
Can you characterize $v$ simply?
What I ...
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An algebraic abstraction of dot and cross product
In a recent physics class, we proved the expansion formula
$$ \vec{\nabla}\times(\vec{A}\times \vec{B}) = \vec{A}(\vec{\nabla}\cdot\vec{B}) + (\vec{B}\cdot\vec{\nabla})\vec{A} -(\vec{B}(\vec{\nabla}\...
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Derivative of cross product w.r.t. a vector
How to compute the derivative of $\vec{a}\times\vec{b}$ w.r.t. $\vec{c}$, all of which are 3D vectors for simplicity? Here $\vec{a}(\vec{c})$ and $\vec{b}(\vec{c})$ are both dependent on $\vec{c}$. I ...
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How to determine the reflex angles in a concave polygon in 3D?
For a concave polygon in 2D, it's easy to use the cross product to determine the reflex angles, which are greater than $180^{\circ}$, but I wonder if there is a simple way to do it in 3D.
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$\hat i\times\hat j$
Very basic question ahead.
It is required to evaluate the cross product of $\hat i$ and $\hat j$, that is, $\hat i\times\hat j$. Knowing that the cross product is anti-commutative, I made sure to ...
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A visual or intuitive proof of $A\times (B\times C)=B(A\cdot C)-C(A\cdot B).$ (an equality in Richard P. Feynman's book)
I am reading Richard P. Feynman's book.
In this book, Feynman wrote the following equality without a proof.
$$\mathbf{A}\times (\mathbf{B}\times\mathbf{C})=\mathbf{B}(\mathbf{A}\cdot\mathbf{C})-\...
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I have read the statement Expectation of two independent variable X and Y E(XY)=0. How to prove it?
I know that if X and Y are independent variables then the expectation value of X and Y i.e $ E(XY)=E(X).E(Y).$ But then the expectation value will be equal to zero only when either of them i.e $E(X) ...
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Why cross product gives area of parallelogram formed by two vectors
Recently we were introduced to the concept of vectors in our class, and we learnt about dot products and cross products.I do know that $a\times b$ yields the area of the parallelogram formed by the ...
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Understanding the cross product of a partial derivative and a vector
I am trying to understand the expression
$$
\frac{\partial(x-y)}{\partial x} \times a,
$$
where $x$, $y$, and $a$ are vectors. I am unsure of how to compute this expression and would appreciate some ...
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Do all bivectors simplify to 2-blades in seven-dimensional space?
The wedge product of two vectors $\vec{v}, \vec{w}\in\mathbb{R}^{n}$ can be defined as an anti-symmetrized tensor product. In three dimensions, there is a correspondence between the wedge product of ...
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Flux through tetrahedron and parametrization of triangular surfaces
I'm struggling with a math problem in my Math for Physics class. We just introduced Gauss's law and now I'm supposed to calculate the flux through a tetrahedron:
Let
$$\vec{A}: \mathbb{R}^3 \space \...
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How to prove the Grassman identity: $(a\times b)\times c=(a\cdot c)b-(a\cdot b) c$ [closed]
I was reading a book and I found this question :
how to prove the Grassman identity : (a × b) × c=(a ⋅ c) b−(a ⋅ b) c
where a,b and c are vectors.
There was a hint that I should begin by expanding the ...
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Geometry of $|A||B| = |A * B| + |A \times B|$
Geometry of $(|A||B|)^2 = |A * B|^2 + |A \times B|^2$ where $*$ represents the dot product
This identity is easy to verify algebraically once we recall that $|A * B| = |A||B|||\cos(\theta)|$ and $|A \...
user1167361
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Can we define this alternate version of the cross product?
I've been playing with the definition of the cross product and am trying to grasp the atomic algebraic assumptions needed to define the unique cross product. I remember seeing a post that was saying ...
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Prove that if $\textbf{g}(t)\parallel \textbf{g}''(t)$ the area of the triangle $OAB$ does not depend on the variable $t$
Let's assume $\textbf{g}(t)$ is a position vector of point $A$, where $t$ is a variable. In addition, $\textbf{g}'(t)$ is the position vector of point $B$. I have to prove that if $\textbf{g}(t)\...
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Where is the magical sign change under change of basis ? Not pseudotensor?
I'm sorry for the long post but the this subject is confusing to me.
Context: On one hand wiki talks about pseudovectors as if they are maps $\Phi:V^k \to V$ on the physical vector space with the ...
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How to determine a vector in $\mathbb R^3$ from its dot and cross products with a given vector
Suppose we have a vector $a$ in $\mathbb R^3$ and an unknown vector $v$, but we know $a \cdot v$ and $a \times v$. Can we find $v$? How?
Sources: Based on Shifrin's Multivariable Mathematics and MIT ...
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Question in pseudovectors
I learnt that under parity transformation a vector
$\vec{A}$ <---(Parity)------> $-\vec{A}$
and a pseudovector can be written as $\vec{c}=\vec{A} $ $\times$ $\vec{B}$ and since A goes negative A ...
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Is it possible to simplify expression with cross product?
So I have an expression
$$
J^{-1} \left( w \times (J w) \right)
$$
where $J$ is a 3x3 invertible matrix and $w$ is 3 element vector and $\times$ is cross product.
Is it the same as
$$
(J^{-1} w) \...
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Cross/dot product identity
In order to prove
$[A\times(B\times C)]\cdot \{[B\times (C\times A)] \times [C\times(A\times B)]\}=0$
With the triple product I got
$[(A\cdot C)B -(A\cdot B)C]\cdot \{[(A\cdot B)C-(B\cdot C)A)] \times ...
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Question about Euler's equation for rigid bodies
I wished to understand a particular case of Euler's equation applied in the following cylindrical body:
where $I_{1,2,3}$ are the moments of inertia. By symmetry, $I_1=I_2=I_T$.
Here I consider that ...
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A problem with cross product of two vectors
Problem:
Let $u$ and $v$ be vectors in $\mathbb{R}^3$. Can $u \times v$ be a non-zero
scalar multiple of $u$?
Answer:
Recall that:
$$ u \times v = (u_2 v_3 - u_3 v_2)i + (u_3v_1 - u_1 v_3)j + (u_1 ...
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Why can the del operator cross product a triple integral be placed inside the triple integral?
Consider the (electric) vector field
$$\pmb{E}(\pmb{r})=k_e\iiint_V \frac{\rho(\pmb{r_s})}{\lVert \pmb{r}-\pmb{r_s} \lVert^2}\frac{\pmb{r}-\pmb{r_s}}{\lVert \pmb{r}-\pmb{r_s} \lVert}d\tau\tag{1}$$
...
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Dot product symbol for divergence, merely a convenience? [duplicate]
As the title says, is it merely a convenience to write the divergence as a dot product? Is there an intuition on the relationship between the geometric interpretation of the divergence and that of the ...
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Spivak's construction of the cross product
I'm having some trouble reproducing Spivak's construction of the cross product (p.83 in his "Calculus on Manifolds").
He restricts it to $\mathbb{R}^n$, and does so by defining for $v_1,\...
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Can a purely mathematical version of the right hand rule be given?
The right hand rule is a common convention for describing orientation of coordinates, used throughout physics. It's also used in the definition of the cross product.
Is it possible to give a purely ...
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Step by step derivation of vector cross-product expression
Equation 1
$$\vec{a} * \vec{b} = (|\vec{a}| | \vec{b}| sin θ)ň $$
Equation 2
$$\vec{a} * \vec{b} = (a_yb_z - a_zb_y)î - (a_xb_z - a_zb_x)ĵ + (a_xb_y - a_yb_x)k̂ $$
I have managed to figure out and ...
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answers
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Manipulation of a multiple cross product expression using skew symmetric matrices
In the derivation of equation (8.23) in the book "Modern Robotics: Mechanics, Planning, and Control" there are some manipulations of the term for the moment contribution of the centripetal ...
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Reference Request: Intuition for Cross Products
I recall reading a blog post a long (~10 years?) ago that gave some intuition for how to reason with cross product identities via infinitesimals, and I cannot find it now for the life of me. Does ...
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What if I want the cross product to have a unit?
This is a strange request, but still.
I encountered this kind of expression:
$$
v_1+v_2+v_1\times v_2
$$
If these were numbers, I would profitably rewrite this as
$$
(1+v_1)\times(1+v_2)-1
$$
but of ...
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Is this a simpler way to orthogonalize bases than Gram-Schmidt?
This is my first post here. I have a question. In my linear algebra course we are learning the Gram-Schmidt process. But it appears to me in a much more intuitive way to do the cross product ...
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Is the 2-tensor equivalent of the cross product incomplete in Exercise 8-2.3 of Ted Shifrin's Multivariable Mathematics?
This is exercise 8-2.3 from Ted Shifrin's Multivariable Mathematics (a good book).
Suppose $\mathbf{v},\mathbf{w}\in\mathbb{R}^{3}.$ Show that
\begin{align*}
dx\left(\mathbf{v}\times\mathbf{w}\right)=...
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Are there different types of zero vectors?
Take the cross product
$\vec{a}×\vec{b}=0\tag{1}$
Where $|\vec{a}|,$$|\vec{b}|$$≠0$ and the angle between $\vec{a}$ and $\vec{b}$ is $0$
Cross product of two vectors can be used to find the vector ...
user1139371
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Prove $(\vec a \times \vec a) \times \vec b = \vec a \times (\vec a \times \vec b)$ if and only if $\vec a = \lambda \vec b$
Prove $(\vec a \times \vec a) \times \vec b = \vec a \times (\vec a \times \vec b)$ if and only if $\vec a = \lambda \vec b$. Here's my attempt. Since I need to prove two directions, I prove it one by ...
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How to prove the following vector quadruple cross product using skew-symmetric representation
in the following expression I'm curious to know is there a way to prove the following quadruple cross product:
$$
\vec{a} \times (\vec{b} \times(\vec{a} \times \vec{b}))=\vec{b} \times (\vec{a} \times(...
0
votes
1
answer
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Mixed product of second-order tensors and vectors
I was studying the angular momentum equation in the continuum case and I encountered this identity. I am not sure how the identity is derived. Could some one supply more details and intermediate step?
...
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Exterior algebra: property of signed area
The wiki on exterior algebras lists of number of properties enjoyed by the signed area, and all of them make sense except for this one:
$A(v + rw, w) = A(v, w)$ for any real number $r$, since adding ...
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Solve vectorial linear system with cross product
Question: Does the system bellow have any solution for $\vec{F}_a$ and $\vec{F}_b$? If yes, how can I find them all?
$$
\begin{cases}\vec{F}_a + \vec{F}_b = \vec{F}_r \\
\vec{p}_a \times \vec{F}_a + \...
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projecting image onto edge of a view frustum for offscreen objects that are not culled
I have a view frustum that works great when looking at stuff from a distance. But for example, when i stand in the middle of a square, my current system struggles to place vertices which are behind me....
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Why the direction of a cross product vector multiplication is considered perpendicular to the area?
I know why cross product is used and how to use it. I also understand that, for doing math we need to consider a direction for the result we have got by doing cross product vector multiplication.
But ...
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Show that $a\times (a\times b)=-(a\bullet a)b$
Let $a$ and $b$ be orthogonal vectors in $\mathbb R^3$
I am trying to show that $a\times (a\times b)=-(a\cdot a)b$ where $\cdot$ is dot product.
We have $a\cdot b=0$. Also I think that $a\times (a\...
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How is $\nabla (u\cdot A) =u\cdot \nabla A+ u\times (\nabla \times A) $?
This was used in the answer here, in the derivation of the Lorentz force law from the Lagrangian. $u$ and $A$ are vectors, the velocity of the particle and the spacetime dependent Magnetic field
As ...
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Can we relate these four determinants?
\begin{equation}
D_1
=\det
\begin{pmatrix}
\alpha_1 &1& \beta_2\\
\alpha_2 &1& \beta_3\\
\alpha_3 &1& \beta_1
\end{pmatrix}
\end{equation}
\begin{equation}
D_2 = \det
\begin{...
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Cross product, Dot product.
I have been struggling with a homework question and I would like to get some help.
The first section of the question it prove the Triple Product Identity:
$ (\vec u \times \vec v)\times \vec w= (\vec ...
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