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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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Skew Symmetric Matrix vs. Cross Product

This might be more of a programming question in truth (at least, I suspect the answer is related to computer programming), but I figured I'd ask here. Why would someone choose to represent a cross ...
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Understanding the Relationship Between Cross Product Components and Differential Forms on a Membrane

I am struggling with a differential geometry and vector calculus concept involving the relationship between the cross-product components and differential forms. The specific context is a response ...
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Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
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Surface area element in 4-dimensions

In Dirac's "General Theory of Relativity" (p. 40) he says "If we take two small contravariant vectors $\xi^\mu$ and $\zeta^\mu$, the element of surface area that they subtend is ...
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Loop integral of $\mathbf{r}\times d\mathbf{r}$ is equal to twice the enclosed area?

I came across the following integral in the literature for a loop $l$ that goes around a surface of area $A$: $$ \oint_l \mathbf{r}\times d\mathbf{r} = 2 A \mathbf{\hat{n}} $$ where $\mathbf{\hat{n}}$ ...
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Tangent Vector, Principal normal vector, Binormal vector, and Torsion

So I'm trying to fully grasp how all these relate. My current understanding is that the tangent vector describes the direction in which the curve is going/curving. Meanwhile, the principal norm is ...
A Student 's user avatar
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Cross product properties.

I just want to know if it's correct: let $\vec{v}=(\vec{a}+\alpha\vec{b})\times(2\vec{a}+\vec{b})$, with $\alpha\in\mathbb{R}$. If $||\vec{a}||=\sqrt{2},||\vec{b}||=1$ and the angle between $\vec{a}$ ...
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What is a space which is the cross product $D^2\times S^1$?

While taking cross product of two one-spheres, $S^1\times S^1$ seems esasy to imagine/identify with a torus $T^2$, I struggle to make a picture of a $D^2\times S^1$, what kind of a space is that in ...
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Finding vector equation of a line

Show that the equation of a straight line passing through the point with position vector $\vec{b}$ and perpendicular to the line $\vec{r}=\vec{a}+\mu \vec{c}$ is of the form $\vec{r}=\vec{b}+\beta \...
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Prove that the cross product is a bilinear form

I want to show that the application $$\phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \quad \text{given by} \quad \phi(u, v)=u \times v$$ It is bilinear In first place, note that \begin{align*} ...
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Interior, cross and outer products between two multivectors?

For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of the interior (or scalar) product $\mathbf u\cdot \mathbf v$, the cross product $\mathbf u\times \mathbf v$ (if ...
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Reasons of computing smallest eigenvalue $R^TR$ instead of singular value

I have problems in understanding why author of this article uses smallest eigenvalue of a cross product matrix instead of a data matrix. I know that $SVD(AA^T)=UD^2U^T$, but I don't know why not ...
Wilk's user avatar
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How do you show that $u+w+v = 0$ given $u \times v = v \times w = w \times u$ [duplicate]

Given that $u+w+v = 0$, I was able to prove that $u \times v = v \times w = w \times u$ by using the anti-commutative property. But I'm struggling a lot with how to approach to prove the converse. ...
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Does cross product depend on the orthogonality of the basis vector [duplicate]

When I learn cross product, I find myself always using orthogonal basis vectors (e.g. $\hat{i}$, $\hat{j}$ and $\hat{k}$). But I am wondering does cross product depend on the orthogonality of the ...
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How to algebraically derive the determinant expression for the cross product [duplicate]

I was reading this post: Deriving formula for cross-product. and I understand the derivation there of the formula and it's easy enough to just manually check that the resulting formula is the same as ...
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Generalize an identity of cross products to $n$ dimensions

For any vectors $x,y,a\in\Bbb R^3$, let \begin{align}f(x,y)&:=x⋅y\\ f(x,y,a)&:=f(x,y)f(a,a) - f(x,a)f(y,a) \end{align} then $$\tag1 f(x,y,a)=(x×a)⋅(y×a)$$ To generalize this to 4D, I found by ...
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Evaluation of surface integral $\iint F ds$; $F(x,y,z)=(x,y,z)$

Let $F(x,y,z)=(x,y,z)$. $\\[10pt]$ Evaluate: $\iint_{S} F ds$; where $S$ is the upper hemisphere of radius $3$, centered at the origin. I defined $\phi(u,v)=(\sqrt{9-v^2}\cos(u),\sqrt{9-v^2}\sin(u),...
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Computing $U \times V$

Problem: Given $$ U = (2,1,3) $$ $$ U = (4,-2,2) $$ find $U \times V$. Answer: \begin{align*} U \times V &= \begin{vmatrix} i & j & k \\ 2 & 1 & 3 \\ 4 & -2 & 2 \end{...
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Cross-product Exercise

I am doing some problems in Spivak's calculus on manifolds, and I was stuck on this one. It says that: If $w_1, ... ,w_{n-1} \in \mathbb{R}^n$, show that $$| w_1 \times ... \times w_{n-1}| = \sqrt{\...
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Determining value of unit vector that satisfies cross product equation

Consider the equation given by: $\langle 0,-1, 0 \rangle$ = $-qV \times \langle -1,0,0 \rangle$, where $\times$ denotes the cross product. I have to find a suitable standard basis vector $V$ that ...
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Intuition for why a 90 degree rotation of a vector about an arbitrary axis can be expressed as 3 90 degree rotations of the vector's projections.

Given a unit vector $\hat{u}$ and a vector $\vec{v}$ perpendicular to $\hat{u}$, we can rotate $\vec{v}$ by 90 degrees around $\hat{u}$ with the cross product $\hat{u} \times \vec{v}$. Since the cross ...
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Prove that the trajectory of $P$ is a circle in $3D$ and find its properties

A point $P=(x,y,z)$ starts off at $P(0)= (x_0, y_0, z_0)$. Its time derivative is given by $ \dfrac{d P}{dt} = a \times P $ where $a \in \mathbb{R}^3$ a unit vector, and $\times$ is the cross product....
c'est pas normale's user avatar
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On vector multiplication [closed]

In this video (check it out, it's worth it), F. Holmér nicely derives the dot and cross product (with some insights into quaternions, the wedge product and much more), just by using ordinary ...
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A question about proof of bac-cab rule

I am trying to prove BAC-CAB rule. However, I am not sure about how can I guarantee that $\gamma=1$ is a solution for each vectors? If my proof is not correct, how should I proceed? I had the ...
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How to prove that the cross product doesn't satisfy any kind of generalized associativity?

It's well known that the cross product in $\mathbb{R}^3$ doesn't obey the associative law of $$ A \times (B \times C) = (A \times B) \times C $$ We can define a "Generalized Associative Law" ...
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Levi-Civita symbol for cross product generalisation, geometric interpretation, orthogonal subspace

Cross Product in $\mathbb{R}^3$ In $\mathbb{R}^3$ one can compute the coordinates of the cross product between $b_2$ and $b_3$ using the Levi-Civita symbol as follows: $$b_{1,i_1} = \sum_{i_2i_3} \...
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Is it true that $\vec{\nabla}\times[(\vec{a}\cdot\vec{\nabla})\vec{a}]=\vec{\nabla}\times[\vec{a}\times(\vec{\nabla}\times\vec{a})]$?

I'm reading some notes on particle physics by a university professor, and after doing calculations I've reached the conclusion that in order for one of his claims to be true, this equality would have ...
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Prove a subset $E$ of $\mathbb{R}^3$ is exactly a plane if there are vectors $u,v,w\in\mathbb{R}^3$ such that $v$ and $w$ are linearly independent and

Prove that a subset $E$ of $\mathbb{R}^3$ is exactly a plane, if there exists vectors $u,v,w \in \mathbb{R}^3$, such that $v$ and $w$ are linearly independent and $$E = u + \mathbb{R}v + \mathbb{R}w$$ ...
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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 ...
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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 ...
Geert-Jan's user avatar
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364 views

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) ...
MURALI K's user avatar
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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 ...
Aniket Harit's user avatar
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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 ...
Maximal Ideal's user avatar
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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 ...
Nour 's user avatar
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1 answer
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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 \...
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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 ...
BENG's user avatar
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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)\...
kauselis3000's user avatar
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2 answers
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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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Identity for a scalar quintuple product?

I find myself needing to cross two pairs of vectors, and cross that result (so a normal of normals) and check whether each of two of the original points are on different sides of the plane it defines: ...
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