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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38 views

Cross product word problems do not make sense to me in textbook [closed]

I'm having trouble following both of these examples.  In 4, I can't figure out what is going on? What is the moment of force? What is the moment of the force p about the center Q of a wheel? What is ...
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Find all functions f(t) such that x = (cost, sint, f(t)) is a plane curve

Okay so I have a question How do we find all function f(t) such that x = (cost, sint, f(t)) is a plane curve I know this means the torsion is 0. So I know that we can find the pieces of the TNB needed ...
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Is there a 'geometric' version of this derivation of the vorticity equation?

Recall that for each vector $\omega\in\mathbb R^3$, there is an anti-symmetric matrix $ [\omega]_\times\in\mathbb R^{3\times 3}$ (and vice-versa) such that $$[\omega]_\times h= \omega\times h.$$ ...
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If $A,B,C$ are collinear, prove $\vec{A}\times \vec{B} + \vec{B}\times \vec{C} + \vec{C}\times \vec{A} =\begin{pmatrix} 0 \\ 0\\ 0 \end{pmatrix}$

Prove that if $A, B$ and $C$ are collinear, then $\overrightarrow{A}\times \overrightarrow{B} + \overrightarrow{B}\times \overrightarrow{C} + \overrightarrow{C}\times \overrightarrow{A} =\begin{...
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1answer
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Dot and cross product of vector valued functions

The given is $$r(t)=\frac{3}{2}t^2\pmb i-4t\pmb j+\frac{1}{2}t^3\pmb k$$ Taking the first two derivatives: $$r^{'}(t)=3t\pmb i-4\pmb j+\frac{3}{2}t^2\pmb k$$ $$r^{''}(t)= 3\pmb i+3t\pmb k$$ Taking the ...
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Why does the defining action of $\mathfrak{so}(3)$ conincide with the adjoint action?

The real Lie algebra $\mathfrak{so}(3)$ consists of infinitesimal rotations, i. e., skew-symmetric operators $\mathbb{R}^3\to\mathbb{R}^3$. Given an orientation on $\mathbb{R}^3$, the vector space of ...
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Inner product, cross product, proof [closed]

$\vec y(t,s)=\vec \alpha(t)+s\vec N(t)$ and $\vec N'(t)=\lambda(t) \vec \alpha'(t)$ is given. I need to show $\langle \vec \alpha',\vec N(t) \times \vec N'(t) \rangle =0$ ( $s$ is a scalar, $\vec N(...
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How to get the 3 points that are collinear in 3 different line equations with a common variable?

So I already have 3 defined equations that give me 3 different lines, two of which are curved. Doesn´t matter how I got them, they are right. They are simply a vector plus another vector multiplied by ...
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Why is the derivative of of the unit tangent vector a cross-product?

Equation for T(t) Equation for curvature Alternate curvature equation Using the equation for T(t) and the equatian for curvature, how does the quotient rule when solving for dT/dt become a cross-...
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How to prove that $(M\mathbf {a} )\times (M\mathbf {b} )=(\det M)M^{-T}(\mathbf {a} \times \mathbf {b} )$? [duplicate]

In computer vision, there are lots of formula came in being with geometry form. And this equation is very about Fundamental matrix in epipole geometry. So, can some one proof this equation? $${\...
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Find a matrix $S$ such that the lie subalgebra $gl_S(n,\mathbb{R})$ is isomorphic to the Lie algebra $\mathbb{R}^3$ where $[x,y] = x \times y$

Let $S$ be an $n \times n$ matrix with entries in field the $\mathbb{R}$ and define: $gl_S(n,F)=\{x \in gl(n,F) : x^tS = -Sx \}$ Find a matrix $S$ such that the lie subalgebra $gl_S(n,\mathbb{R})$ is ...
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How do I use cross products to find the area of the quadrilateral in the $𝑥𝑦$-plane defined by $(0,0), (1,−1), (3,1)$ and $(2,8)$?

How do I use cross products to find the area of the quadrilateral in the $𝑥𝑦$-plane defined by $(0,0), (1,−1), (3,1)$ and $(2,8)$? So what I first do is find two vectors. Gonna use (0,0) as ...
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1answer
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Solved! Cross product in parallelogram

The area of the parallelogram $ABCD$ is equal to: $$\text{Area} = \left|\vec{AB}\times\vec{AD}\right|\text.$$ Is it the same for $$\text{Area} = \left|\vec{AC}\times\vec{AD}\right|\text?$$ Because ...
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Question Regarding Cross Product Identity

I am trying to understand how $$\vec{r} \times(\alpha \times \vec{r}) = $$ $$\alpha(r \cdot r) - r(r \cdot \alpha)$$ The context of this question is shown in the image below. Thank you.
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Find the values for $u$ and $v$

Given $u = (2, 4, −3)$,and $v = (3, −1, 7)$. Find $u\cdot v$ and $u\times v$. What's the difference between $u\cdot v$ and $u\times v$? $u\cdot v$ is the dot rule and what is $u\times v$?
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How to simplify expression for volume of tetrahedron using vectors

Below is the question I have been working with: And here is the solution: Now, I’m stuck with part c. I’ve tried searching online but noone seems to be focusing on the problem that keeps me from ...
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$P\cdot (Q \times P)$ where $P$ and $Q$ are vectors

The answer is zero, but why? my theory is that $P.Q$ is a scalar product, you cant do cross product between the remaining vector and scalar but it was written in the answer that the cross product of ...
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How do you derive the formula $d= \frac {|a \times b| }{|a|}$ to find the shortest distance between 2 vectors?

Distance from point $P$ (not on $L$) to line $L$ (that passes through $Q$ and $R$) is $$d=\frac{|\vec{a}\times \vec{b}|}{|\vec{a}|}$$ where $\vec{a}=\vec{QR}$ and $\vec{b}=\vec{QP}$ Find the distance ...
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Given $\vec{v_1} \times \vec{v_2}$ and $\vec{v_1} \times \vec{v_3}$, find $\vec{v_2} \times \vec{v_3}$

Is it possible to get the cross product of two vectors $\vec{v_2}$ and $\vec{v_3}$, given the result of each cross product with a common vector $\vec{v_1}$? e.g. $$ \vec{v_i} \in \mathbf{R}^2_+ \\ \...
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How can I prove $[r][\omega]^{2}r = -[\omega][r]^2\omega$?

A derivation I am reading from a book requires me to prove $[r][\omega]^{2}r = -[\omega][r]^2\omega$ . Now this was part of a larger derivation and hence the book skipped a few intermediary steps and ...
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Finding a general equation of a plane

Let $x = [3, 4, 2], y = [2, −1, 3],$ and $z = [−1, 2, 1]$. Give a general equation of the plane $P$ in $\mathbb{R}^3$ which passes through the point $[1, −2, 2]$ and has direction vectors $x$ and $y$. ...
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If a.i=4, then ,what is the value of (axj).(2j-3k) , where a is a vector

This is a question I saw in a question paper of a competitive exam but I was unable to solve it. Can anyone please assist me with any sort of hint to solve this problem and any type of explanation if ...
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1answer
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Deriving Parallel and perpendicular vectors from triple vector product

How would one go about resolving the vector $\vec{p}$ into parallel and perpendicular vectors to the given vector $\vec{w}$ By considering - $\vec{w}\times(\vec{p}\times\vec{w})$ So far I have ...
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1answer
50 views

2 Cross Products?

Usually, if we want to find the cross product of 2 vectors $\vec{b}$ and $\vec{c}$, we want to find the vector which is perpendicular to both of them. Let's say the cross product of $\vec{b}$ and $\...
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Properties of cross product

Let $u=[3,−2,1]$, $v=[1,1,1]$ and $w=[2,−2,0]$. The parallelepiped formed by $u$ and $v$ and $k(u × v)$ will have the same volume as the parallelepiped formed by $u,v$ and $w$ if $k=±\frac{1}{n}.$ ...
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Suppose $t, u, v, w \in \mathbb{R}^3$. If $(t \times u) \times (v \times w) = 0$, are $t,u,v,w$ on the same plane?

Suppose $t, u, v, w \in \mathbb{R}^3$. If $(t \times u) \times (v \times w) = 0$, are $t,u,v,w$ on the same plane? My Answer No. Let $p_1$ be the plane described by $x + y + z = 3$ and $p_2$ the ...
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Proof of cross product in orthogonal transformation (using levi civita)

I am trying the following proof. If objects X and Y transform as vectors under orthogonal transformation, then the cross product of X and Y also transforms as vector. So I try this way; Let $$x_{j}^{'...
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1answer
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Integral definition of curl

My book, Mathematical Methods for Physics and Engineering - K. F. Riley, explains that the 'Integral definition of curl' is:- $$\nabla\times a = \lim_{V\to0}(\frac{1}{V}\int_S dS\times a) $$ and then, ...
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Proof of triple product [duplicate]

Consider this proof of the triple product identity. In this proof, they state that $\lambda $ does not depened on any choice of three vectors $A,B,C$. I don't understand why is that true? I'll ...
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Triple product proof [duplicate]

I am trying to prove the identity $a\times(b\times c)=\langle a,c\rangle b-\langle a,b\rangle c$ Like other proofs (such as this one) , I have managed to find that $a\times(b\times c)=\lambda\langle a,...
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Matrix-vector multiplication/cross product problem

How can I generally solve equations of the form $\mathbf{A} \mathbf{w} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \times \mathbf{w}$ for the matrix $\mathbf{A},$ where $\mathbf{w}$ can be any vector? ...
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Division using cross vector

Given $4$ vectors $\vec a$ and $\vec b$, $\vec c$, $\vec d$, if $$\vec u=\vec a \times \vec b= \hat n \ ab \sin θ \tag 1$$ where $\hat n$ is a normal unit vector of both $\vec a$ and $\vec b$, and $θ$ ...
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How to simplify Kronecker delta with einstein summation?

I am trying to proof a vector identity. I have to prove the following; I am bit confused how to simplify the following part.. $$\delta_{il} \delta_{jm} x_{j}y_{l}z_{m}$$ Any input is appreciated.
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Is this function, containing a cross product continuous?

Is the following function continuous ? f : $\mathbb{R^3}\times \mathbb{R^3} \rightarrow \mathbb{R^3} $ $x,y \mapsto x \times \frac{y-x}{|y-x|}$ for $x \neq y$, else $0$. Now I would have argued that ...
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Is there a good (non-calculational) reason for the formula $|v \times w|^2 + (v \cdot w)^2 = (|v||w|)^2$?

If $v$ and $w$ are 3D vectors, we have the formula $$|v \times w|^2 + (v \cdot w)^2 = (|v||w|)^2.$$ This formula is used to give the magnitude formula $|v \times w| = |v| |w| \sin(\theta)$. But the ...
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Ray intersection with cylinder of arbitrary rotation

I'm working on writing an algorithm for a the distance of a ray intersection with a cylinder (t), where the cylinder is of arbitrary rotation. Using this website as inspiration, I know I can find the ...
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1answer
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Deriving formula for cross-product.

It is given on pg. #106, 107 in the book by: Thomas Banchoff, John Wermer; titled: Linear Algebra Through Geometry, second edn.. Consider a system of two equations in three unknowns: $$a_1x_1 + a_2x_2 ...
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What does orthonomality mean?

According to me two vectors are orthogonal when they perpendicular so that their inner product is zero. But what do we mean that two functions are orthogonal ? Or how to derive this? $$\langle f,g \...
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Has the operator $\frac{u \times v}{u^T v}$ a name?

The operator $\frac{u \times v}{u^T v}$ gives a vector in the direction of the cross product, with magnitude equal to the tangent of the angle between $u$ and $v$. Does this thing have a name?
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Recovering three dimensional vectors after projection and cross product

Suppose $e_i \in \mathbb{R}^3$, $1\leq i \leq 3$ with $\Vert e_i \Vert=1$. Suppose $u,v \in \mathbb{R}^3$, $u^T v=0$, $e_i^T u \neq 0$, $\Vert u \Vert =1$. Suppose $k\in \mathbb{R}$. Define the ...
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How to rearrange vectors in cross product? [duplicate]

I have an equation: $\vec{v}=\vec{w}\times \vec{r}$ How to I separate the $\vec{w}$ and write it in terms of $\vec{v}$ and $\vec{r}$? I tried to re-arrange this equation so that I could find the ...
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Rank of a particularly structured matrix

Suppose $e_1,e_2,e_3 \in \mathbb{R}^{3\times1}$ is an orthonormal basis for $\mathbb{R}^3$, and $f_1,f_2,f_3\in \mathbb{R}^{3\times1}$ is another orthonormal basis for $\mathbb{R}^3$. Suppose $a_i,b_i,...
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Connection between cross product and determinant

This may be a stupid question to some, but when i calculate a cross product of two vectors. For example the first coordinate of the solution. I put my finger on the first line, then i calculate ...
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1answer
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Find vector with specific angles

$ v = [{1, 1, 0}] $ and $ w = [-1, 0, 1] $. Find such vector $ u \in \langle v, v \times w \rangle $, that: $$ 1) \space \space \alpha(u, v) = \frac{\pi}{3} \\ 2) \space \space \alpha(u, v \times w) =...
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1answer
76 views

Volume of parallelepiped given three parallel planes

Prove that the volume of the parallelepiped formed by the planes: $a_i x + b_i y + c_i z = p_i$, $a_i x + b_i y + c_i z = q_i$ , $i= \{1,2,3 \}$ is $$ \left|\frac{\prod_{i=1}^{3} (p_i - q_i)}{\begin{...
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1answer
27 views

Confusion regarding a number in a cross product problem.

I'm practicing cross product and everything is going fine until I reach a certain problem. The vectors I'm given are <5, -3, 3> and <4,-6, 0> respectively. My answer is <18,-12, 42>. ...
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2answers
57 views

Question on vector cross product.

Show that $\big((\mathbf{u} \times (\mathbf{u}\times \mathbf{v})) \times \mathbf{v}\big) \times (\mathbf{u} \times \mathbf{v})=0$. From wolfram, it gives zero but there's no details. How to prove ...
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1answer
37 views

Vectors and Cross Product in 3D

I set the vector V as (a,b,c). I know that you have to multiply out the two vectors, so that is what I did. I then got multiple equations which I used to find the values of a, b, and c. The answer ...
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1answer
49 views

Geometric interpretation of the scalar triple product

The Wikipedia article for scalar triple product says the following: Geometrically, the scalar triple product $$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})$$ is the (signed) volume of the ...
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1answer
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Cross product and projection [closed]

$\textbf{Problem:}$ Let $v_1,...,v_{n-2} \in \mathbb{R}^n$ with $\{v_1,...,v_{n-2}\}$ is linearly independent. Let $B = (v_1,...,v_{n-2})$ and $C = \text{Col}(B)$. If $x \in \mathbb{R}^n$, show that $$...

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