For questions on cross products.

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2
votes
3answers
408 views

showing / proving curl identity $\nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$

OK, I have to show the following: $$ \nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$$ This should be pretty easy, but I wanted to be sure I was doing this correctly. I set up the matrix: ...
0
votes
3answers
51 views

Help proving that $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = -\|\vec{a}\|^2\vec{a}\cdot\vec{b}\times\vec{c}$

This is problem 13 from Chapter 13, Section 14 of Apostol's Calculus Vol 1. I need to prove or disprove the formula $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = ...
2
votes
1answer
42 views

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3, |V| = 2$ and $U\cdot V > 0$, then what is the angle between $U$ and $V$?

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3$, $|V| = 2$ and $U \cdot V > 0$, then what is the angle between $U$ and $V$? Is this just dot product? $U \cdot V = |U| |V| \cos(\theta)$? Solve ...
2
votes
1answer
47 views

Unbiased estimate of cross-product for unbiased vector

Let $g$ be an unbiased estimate of a vector $G$. Can $g$ be used to find an unbiased estimate of the cross product $GG'$? I'm stuck because naively using $gg'$ is a biased estimator, with the ...
7
votes
1answer
226 views

Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
1
vote
1answer
121 views

Multiplication of cross product with Levi-Civita symbol

I want to show following property for the scalar product $$ \left( \vec \omega \times \vec y,\vec \omega \times \vec y\right)=\left( \vec \omega,\vec y\times \left(\vec \omega \times \vec ...
0
votes
1answer
994 views

How to prove this vector identity

How do i prove this vector identity ? $$(\vec a \times \vec b)\times \vec c=(\vec a \cdot\vec c)\vec b - (\vec b\cdot\vec c)\vec a$$
3
votes
2answers
60 views

Are these two rotation matrices related?

There are two $3\times3$ matrices $A$ and $B.$ Both represent a rotation in 3D space. $A$ and $B$ are given as follows where $a,b,c$ are column vectors. $A = [\begin{array}{ccc}a & b & c \\ ...
1
vote
1answer
61 views

Bio-Savart Law [Doubt about Cross Product in the equation]

In my physics textbook, Bio-Savart Law is written as: $$\vec{B} = \frac{K\,i\, d\vec{s} \times \vec{r}}{ 4 \pi \, r^2}$$ $K$: constant And, when the cross-product is made, the result is: $$B = ...
2
votes
2answers
65 views

Possible to solve this coupled system of vector equations?

Let $\gamma, \omega, c$ be positive constants, let $\mathbf{Q}_{a}$ and $\mathbf{Q}_{b}$ be three-dimensional vectors, and let $\mathbf{B}(\mathbf{r})=\mathbf{B}(x,y,z)$ be a vector field. Let ...
3
votes
1answer
49 views

Proof that $\mathbf{R}[\omega]_\times\mathbf{R} = [\mathbf{R}\omega]_\times$

I have to prove that $$\mathbf{R}[\omega]_\times\mathbf{R}^\mathrm{T} = [\mathbf{R}\omega]_\times$$ Herein $\omega$ is a vector with elements. The notation $[\mathbf{a}]_\times$ is a conversion of ...
1
vote
1answer
88 views

Determining $u=v \times w$ using the cross product

Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula. What are the possible angles $x$ between two unit vectors ...
5
votes
4answers
2k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
2
votes
1answer
150 views

Multiplying vectors (answered own question)

I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ...
0
votes
1answer
134 views

Category-theoretic cross product and set-theoretic cross product

I recently proved as an exercise the associativity of cross product as defined in category theory. But in set theory, cross product is not associative. It seems intuitive to me that cross should be ...
1
vote
1answer
78 views

Find closest vector to A which is perpendicular to B

To start, I would like to apologize if the answer to my question was easily googled, I am quite new to this and googling "Find closest vector to A which is perpendicular to B" gave me no results. My ...
5
votes
1answer
360 views

Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
0
votes
1answer
87 views

Cross product as factor in dot product [duplicate]

Given there are two vectors $w,v$ with $||w||=4$ , $||v||=1$ and $\phi=\frac{2\pi}{3}$ How do you transform the following expression into a form in which it can be computed with the given ...
0
votes
2answers
68 views

Computing cross product using norm and angle

Sorry for the weird title, if someone finds a better title for my problem be my guest to edit it ;) For $\mathbf{v,w} $ in R³ with $\mathbf{||v||=1 ;||w||=4; \theta =\frac{2\pi}{3}}$ Solve ...
1
vote
1answer
43 views

Find vectors vertical to given vectors with certain length

Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1 Every vector $\mathbf{x'}$ that is to be found must meet these ...
1
vote
3answers
48 views

Reordering vector product

If I have vectors $a, b, c \in \mathbb{R}^3$, and if we have e.g. $a = b\times c$, is there any way to express $b$ in terms of the other two?
3
votes
2answers
170 views

Help over the proof of triple vector product identity

For all vectors $\bf{x}$, $\bf{y}$ and $\bf{z}$, $$\bf{x}\times(\bf{y}\times\bf{z})=(\bf{x}\cdot\bf{z})\bf{y}-(\bf{x}\cdot\bf{y})\bf{z}$$ The proof goes as follows: We may suppose that $\bf{y}$ ...
4
votes
1answer
429 views

Interpretation of eigenvectors of cross product

If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ ...
4
votes
1answer
87 views

How to solve cross-products including matrices?

I'm a programmer and I'm doing a whitebalance-transformation in RGB colorspace. This should work with this transformation matrix that I've found in literature: $$ \begin{pmatrix} R \\ G \\ B ...
3
votes
3answers
150 views

Cross product- square

I recently saw the following expression somewhere- $$\frac{1}{2} \left\| \frac{\vec{u}}{9} \times \frac{\vec{u} + \vec{v}}{9} \right\| + \frac{1}{2} \left\| \frac{\vec{u} + \vec{v}}{9}\times ...
2
votes
1answer
62 views

Cross product, ortonormal basis

Could you explain to me why for $\{i, \ j, \ k\}$ an orthonormal basis of $\mathbb{R}^3$ we have $i \times j =k, \ \ j \times k = i, \ \ k \times i =j$? Thank you.
1
vote
1answer
42 views

Comparing a geometric definition of cross product to the “usual” one

Could you help me with my little problem? Given this definition of cross product: 1) $a \times b$ is perpendicular to $a$ and $b$, whenever $ a,b$ are linearly independent 2) basis $a, \ b, \ a ...
1
vote
1answer
92 views

Solution set to cross product

If $\vec a,\vec b \in \mathbb{R}^3$ with $|\vec a|\ne0$ show that the equation $\vec a \times \vec u =\vec b$ has a solution if and only if $a \cdot b = 0$ and find all the solutions in this case. ...
1
vote
2answers
93 views

Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is maximal, and then a unit vector $w$ s.t $|(u\times v)\cdot w|$ is minimal

This is a similar question to the one I have posted before. The problem is as in the title: Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is maximal, and then a unit vector $w$ s.t ...
-3
votes
2answers
145 views

Why cross product's formulas defined in this way? [closed]

Why cross product's formulas defined in this way? When mathematicians need to define cross product?
2
votes
1answer
139 views

Cross Product for functions

So functions are just uncountabley-infinite dimensional vectors, and as such there's a nice generalization of the inner product between two functions (the integral of their product). Is their a ...
1
vote
3answers
5k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.
1
vote
1answer
126 views

Test of handedness

I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors. if (v1 x v2) . v3 > 0 then it's right-handed, while if ...
1
vote
2answers
165 views

Vectors and Cross Product

I have these two questions regarding the Cross Product. 1.) You are looking down at a map. A vector $u$ with $|u| = 3$ points north and a vector $v$ with $|v| = 10$ points northeast. What is $|u ...
1
vote
1answer
300 views

How to generate an ordered list of vertices of a cube from a face and a normal vector

Consider a cube with faces we'll call "left", "right", "front", "back", "top" and "bottom". The cube can be described by $0 \le x,y,z \le 1$. To name the faces, we'll say $x$ extends to the right, ...
0
votes
1answer
349 views

The Darboux vector is defined by $D = \tau T + \kappa B$. Show that $T' = D \times T$

The Darboux Vector is defined as $D = \tau T + \kappa B$. Show that for a unit speed curve $$T' = D \times T \hspace{1cm} ... $$ Here, the $...$ represents the fact that there are a few ...
8
votes
1answer
154 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
0
votes
1answer
96 views

Special Case of Lie-algebra

Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have ...
-1
votes
2answers
523 views

Find the sine sign given a pair of 3D vectors

I want to find the exact sine between two vectors in 3-dimensional space. Data: $x$: vector $y$: vector $z = \Vert x \times y \Vert$ I have tried this: $$\sin \alpha = \frac{\Vert z\Vert}{( ...
4
votes
3answers
259 views

Rotational invariance of cross product

Hi guys I'm looking for a proof that $ ( Ra \times Rb ) = R ( a \times b ) $ where $\times$ is the three-dimensional cross product, and $R$ is a rotational matrix ( $\det R = 1$ and $R^T R = I$ ) ...
2
votes
0answers
190 views

Cross product in higher than 3 dimensions

As I understand it, to get an $n$-dimensional cross product, you need $n-1$ vectors of dimension $n$. However my lecture notes are quite miss leading in the fact that they suggest this isn't always ...
1
vote
2answers
91 views

What does the symbol $\Delta$ stands for?

While studying Landau-Lifshitz equation following term appears, $-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$ In above equation m is a vector quantity. It will be great if someone can ...
1
vote
1answer
67 views

Vector product proof

Prove that if $$a=b \times c$$ $$b=c \times a$$ $$c=a \times b$$ then $a \perp b$, $a \perp c$, $b \perp c$, and $|a|=|b|=|c|=1$
0
votes
1answer
211 views

Cross product proof

Three vectors $a$, $b$, and $c$ are given. Prove that if $a \perp b$, $a \perp c$, $b \perp c$, and $|a|=|b|=|c|=1$, then $$a=b \times c$$ $$b=c \times a$$ $$c=a \times b$$
2
votes
3answers
7k views

What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
0
votes
1answer
178 views

Cross product as result of projections

The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of ...
1
vote
1answer
178 views

proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$

I encounter this triple product property in wikipedia But I can't find proof for $$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$ The RHS cross ...
1
vote
1answer
226 views

Cross product and inverse of a matrix

I would like to show that $\left(\begin{array}{ccc} 1 & s & s^2 \\ 1 & t & t^2 \\ 1 & u & u^2 \end{array}\right)$ has an inverse provided $s$, $t$ and $u$ are distinct. I ...
0
votes
1answer
54 views

Derivative of a vector function

Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus. Find the derivative of the vector function: ...
0
votes
1answer
43 views

problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$

$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$ can anyone expand on how the final answer is derived? I try to expand but ended up scratching my head. the best I can ...