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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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Is a function on a product measurable space measurable iff it is componentwise measurable?

Let $(\Omega_1, \mathcal{G}_1), (\Omega_2, \mathcal{G}_2), (\Omega, \mathcal{G})$ be measurable spaces and let $$(\Omega_1 \times \Omega_2, \mathcal{G}_1 \otimes \mathcal{G_2}) $$ the canonical ...
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Find distance between a line and the origin [on hold]

The given line is defined as following: $$\text{plane 1: } x+y+z = 6$$ $$\text{plane 2: } 2x - y - 5z = -5$$ What is the distance between the line and the origin?
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What's the point of the cross product?

I don't understand the motivation behind defining cross products the way they're defined. Given two vectors $\vec{A}$ and $\vec{B}$ in $\mathbb{R^3}$, I can find a third vector $\vec{C}$ such that $\...
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Resolving Cross Product Ambiguity Algebraically

Say we have an orthonormal basis in $\mathbb R^3$ $\{A, B, C\}$. Then when taking the cross product of any 2 of these, we know it is equal to either the third basis element, or $-1$ times that element....
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Cross product in polar coordinates

$$\vec{r}\times m(\dot{r}\hat{r}+r\dot{\theta}\hat{\theta})=mr\hat{r}\times (\dot{r}\hat{r}+r\dot{\theta}\hat{\theta})=mr^2\dot{\theta}(\hat{r}\times\hat{\theta})$$ In the first equation I know we ...
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2answers
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Prove that if vectors are independent then their cross product is not $0$

Let $V_1, V_2, ... V_{n-1} \in R^n$ be independent vectors. I want to prove that their cross product $V_1 \times V_2 \times ...\times V_{n-1} \ne 0$. I know that the cross product is equal to the ...
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Levi-Civita symbol - cross product - determinant notation

In this article wiki appears $e_1,e_2,e_3$ in the determinant representing a cross product but it is never defined anywhere - is this some special cross product where $e_i$ can be every thing - an ...
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1answer
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Do all elements belonging to kernel of a linear transformation represented by a skew symmetric matrix NOT belong to its range?

I have a linear transformation represented by a skew symmetric matrix $S(\vec b(t))$ of rank $2$, which in my context, is the cross-product matrix of magnetic field $\vec b(t)$. I have explained what ...
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Vector Calculus - Evaluating $\nabla \times \mathbf{E} = -\frac{1}{c} \partial_t \mathbf{B}$

For the life of me, I cannot remember how to solve equations similar to the cross product equations in Maxwell's equations. I haven't used vector calculus of this level in quite some time and could ...
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Cross Production - Relationship of the area of the parallelogram to the intersection of two 3D lines/vectors

Cross product is defined as the area of the parallelogram that two lines create. Simply put if we have two lines, A and B that create an angle $\theta$ between themselves, the cross product of this ...
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Value of [a b c] from (a×(b×c)).(a×c)?

If a and c are unit vectors at an angle $\pi/3$ with each other and (a×(b×c)).(a×c)=5, then what is the value of [a b c]? I just know the basic meaning of what are vector and scalar triple product. ...
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Deriving a formula for area of a triangle using vector cross product

Using the vector cross product, how would I derive a formula for the area of a triangle with vertices: $$\\(x_0, y_0, z_0)\\(x_1, y_1, z_1)\\(x_2, y_2, z_2) $$ in terms of only $x_0, y_0, z_0, x_1, ...
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1answer
29 views

Rigid body kinetic energy extra term

I am looking at the derivation of the kinetic energy of a rigid body, with an initial velocity: $v = v_{com} + w \times r$ I start with: $T = \int \frac12 v \cdot v \ dm=\int \frac12(v_{com} + w \...
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scalar triple product for continuous functions

How is the scalar triple product for continuous functions, rather than vectors? Can it also be written using the integral of the product of three continuous functions?
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What is the Künneth formula for complete varieties?

I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times ...
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Proving $(\bf x\times y\cdot N)\ z+(y\times z\cdot N)\ x+(z\times x \cdot N)\ y= 0$ when $\bf x,y,z$ are coplanar and $\bf N$ is a unit normal vector

Prove that if $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{R}^3$ are coplanar vectors and $\mathbf{N}$ is a unit normal vector to the plane then $$(\mathbf{x}\times\mathbf{y} \cdot \mathbf{N})\ \...
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Is this possible in vector calculus?

$$(\boldsymbol{\nabla}\alpha)\wedge(\boldsymbol{\nabla} \wedge \boldsymbol{x} )$$ In all the examples in lecture, it has always been a $$\boldsymbol{\nabla}$$ on the left hand side. Does this give ...
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Area of parrelologram with verticesA=(−1,2,4), B=(0,4,7) C=(1,1,3) , and D=(2,3,6)? [closed]

Dont understand how to do this. which two vectors do I multiply?
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2answers
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Cross Product definition

Why is the cross product defined as the area of the parallelogram created by making the vectors $\vec{a}$ and $\vec{b}$ it’s sides, multiplied by the unit vector $\vec{n}$ which is orthogonal to $\vec{...
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Find a plane that goes through three given points

The Question Data is collected on a person’s income (thousands of dollars), their age, and the value of their home (thousands of dollars). We would like to predict home value, H, as a function of ...
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Unit tangent, principal normal and their cross product form an orthonormal basis for $\mathbb{R}^3$

Suppose $\mathbf{T}(s)$ is the unit tangent vector to a curve parametrized by arclength, and $\mathbf{N}(s) = \mathbf{T}'(s)/||\mathbf{T}'(s)||$ assuming $\mathbf{T}(s) \ne \mathbf{0}$. Define the ...
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2answers
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Higher dimensional cross product equivalent

I'm working on a computer vision script for high dimensions that is highly reliant on the cross product in 3D, but as far as I know, it is only formally defined in 3D and 7D. However, experimentally, ...
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How to deterministically pick a vector that is guaranteed to be non-parallel to the given one?

I have a unit vector $u = (x1, y1, z1)$ in $R^3$, given $(x1, y1, z1)$ are known rational numbers. I need to deterministically pick a unit vector $v = (x2, y2, z2)$, such that $v$ and $u$ are not ...
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Linear algebra: cross product property

I am probably a bit lost but I found the following 'cross-product' property in a paper : $(Ma) \times (Mb) = \frac{1}{det([a,b]^{-1})} ( M_1 \times M_2 ) $ With $M$ a $3\times2$ Matrix, $a$ and $b$ ...
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Time derivative of a matrix by using “cross product” (skew-symmetric matrix)

I found a strange way to do the time derivative of a matrix (see eq. 3.159 please). Can you help me to understand why $\dot{I} = \omega \times I - I \times \omega $? I think that $\omega$ is written ...
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1answer
24 views

Collecting cross product inside a matrix

If I have a matrix represented by three column vectors like the one below, can I collect the cross product as I did below? $$ [ \overrightarrow{a} \wedge \overrightarrow{b} \quad \overrightarrow{a} \...
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1answer
21 views

Trouble Finding a Position Function Given a Dot Product & Initial Value

I tried taking the dot product of r(0) and r'(0) and setting that equal to e^0 = 1 and found that <1,0,0> DOT < a,b,c > = 1 therefore a = 1. But I'm not sure how to proceed from here.
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Determine if points P,Q and R are collinear, and if not, find a vector normal to the plane containing them.

Determine if points $P, Q$ and $R$ are collinear, and if not, find a vector normal to the plane containing them. I've never done a collinear problem. There are three sets of points for $P, Q,$ and $R$...
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Tensor cross product proof: $Sa \times Sb = \det(S)S^{-T}(a \times b)$ [duplicate]

I need to prove $$Sa \times Sb = \det(S)S^{-T}(a \times b),$$ given that $a$ and $b$ are vectors and $S$ is a second-order (rank $2$) tensor. I have the hint: vectors $u=v$ iff $u\cdot a=v\cdot ...
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1answer
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Cross Product Calculation Question Given Orthogonality Conditions

Suppose that vectors $u$, $v$, $w$ are mutually orthogonal. Compute $(u \times v) \times w$ and $u \times (v \times w)$ Wouldn't we simply multiply $u$ and $v$ and $w$ all together? Or is there a ...
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Vector Mathematics

I urgently need help correcting a multivariable calculus exam involving vectors. The questions are listed below, along with the answers that I put: 1) Let $P= (1, 0, 1)$, $Q = (4, 1, 9)$, and $R= (3, ...
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Equation of a Plane Perpendicular to Another Plane and Containing a Line

Question involves: Find an equation of the plane which contains the line of intersection of the planes x − z = 1 and y + z = 3, and is perpendicular to the plane x + y − 2z = 1. So far I took n1 as ...
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1answer
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What are the possible angles (theta) between two unit vectors $e$ and $f$ if $|e\times f| = 1/2$?

This is all I was able to muster so far. Our professor didn't show us what to do in this situation of dot products. What next step could I take?
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What does it mean for the cross product to be invariant under orthogonal transformation?

This article claims $Ax \times Ay = \det(A)(x\times y)$, but using the vectors $(3,2,1)$ and $(5,4,6)$ and the orthogonal transformation with determinant $1$: $$\left(\begin{matrix} \frac{1}{2} & ...
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2answers
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Quaternion product of three vectors: meaning of vector part?

$\newcommand{\i}{\mathbf{i}} \newcommand{\j}{\mathbf{j}} \newcommand{\k}{\mathbf{k}} \newcommand{\a}{\mathbf{a}} \newcommand{\b}{\mathbf{b}} \newcommand{\c}{\mathbf{c}} \newcommand{\R}{\mathbb{R}}$If ...
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1answer
88 views

Derivative of cross product

Let $f:V_1\times\dots\times V_N \to W$ be multilinear. Then $f$ s differentiable and $$df(a_1,\dots,a_n)(h_1+\dots+h_n)=f(h_1,a_2,\dots,a_n)+f(a_1,h_2,a_3,\dots,a_n)+f(a_1,\dots,a_{n-1},h_n)\tag{1}$$ ...
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4answers
434 views

Complete $\{(2,3,1),(1,4,3)\}$ to a basis of $\Bbb R^3$

Having the following set: $C = \{(2,3,1),(1,4,3)\}$ I want to be able to generate $V = \mathbb{R}^3$ Since $V = \mathbb{R}^3$ has dimension $3$, $C$ needs $3$ elements. However, how do I find the ...
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Proofing the derivative of Rodigruez's Rotation Formula equals the formula for relating the linear velocity of a point to the angular velocity.

I want to proof that the derivative of Rodriguez's Rotation Formula equals the formula for relating the linear velocity of a point on a moving body to the angular velocity (following the footsteps of ...
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A pyramid $OABC$ has vectors $\vec{OA}=a$, $\vec{OB}=b$ and $\vec{OC}=c$.

A pyramid $OABC$ has vectors $\vec{OA}=a$, $\vec{OB}=b$ and $\vec{OC}=c$. The vectors $v_1,v_2,v_3$ and $v_4$ are perpendicular to each of the faces of and of magnitude equal to the area of the ...
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Distributive property of a matrix on a cross product

Let $\vec x, \vec y \in \mathbb{R}^3$ and $\bf A $ be a $3 \times 3$ real matrix. Under what conditions does $\bf A$ distribute over a cross product: $$ \mathbf{A} (\vec x \times \vec y) = (\mathbf{A}...
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Cross Product in 3D Cylindrical or Spherical coordinates

When i take a cross product of two vectors (Cylinder base in x,y plane) for example d(phi)cross d(z), how do i know if the resultant Vector protrudes out of the page or goes inside? I know the right ...
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What is the “cross product” of $n-1$ vectors in $\mathbb{R}^n$ called?

Is there a name for the following determinant ($v_1, ... v_{n-1}$ are vectors in $\mathbb{R}^n$)? $$\det\begin{pmatrix} \mathbf{e_1}&\cdots&\mathbf{e_n}\\ (v_1)_1&\cdots&(v_1)_n\\ \...
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3answers
55 views

Can't we factor out a constant in the cross product?

I have the vectors $A=a\hat e_x$ and $B=a\hat e_y$, so $$ A\times B = \begin{vmatrix} \hat e_x &\hat e_y & \hat e_z \\ a & 0 & 0\\ 0 & a &0 \end{vmatrix}=\hat za^2 $$ Q1: But ...
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Four dimensional cross product of THREE vectors

There are many MSE posts about how to define a cross product in $\mathbb{R^4}$. It is impossible to define a cross product of two vectors in $\mathbb{R^4}$, since there are infinitely many directions ...
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How to show $\unicode{x222F} \dfrac{\hat{r} \times \vec{dS}}{r^2}=0$

I can do the following derivation using solid angle: $$\unicode{x222F} \dfrac{\hat{r} \cdot \vec{dS}}{r^2} =\unicode{x222F} \dfrac{dS \cos\alpha}{r^2} =\int^{2\pi}_0 \int^\pi_0 \sin\theta\ d\theta\ d\...
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1answer
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Double cross product in 2D

Hello I have a question about a double cross product, appearing in centrifugal force \begin{align*} \mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}] \, ....
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Developing a cross product of tensors within integrals

I read in a book the following unproven statement: $\int_{s} u\times A n \, ds = \int_v ( u\times \nabla\cdot A + \mathcal{E}: A^T ) \, dv$ with a: 1st order tensor, n: normal vector of s, A: 2nd ...
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1answer
69 views

Is the length of the multilinear cross product given by this formula? If not, what is the correct formula?

This is an attempt to make this question more specific. We can compute the length of the cross product in $\mathbb{R}^3$ using the formula: $$|x\times y| = \sqrt{|x|^2|y|^2 - |x\cdot y|^2}$$ This ...
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1answer
33 views

Should dot product must be applied on values of same scale

I have points in $n$-dimensions. I want to find the points which lie on one side of the plane and other lies on the second side and i'm trying to do this with the help of dot-product. Suppose i ...
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Mixed product in tetraeder

We have a correct tetraeder ABCD. On lines AB, AC, BD, CD are points P, Q, R, S, soo that: |AP|/|AB| = 2/3 |AQ|/|AC| = 3/4 |BR|/|BD| = 3/4 |CS|/|CD| = x For what value x are points coplanar? What ...