For questions on cross products.

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2
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5answers
93 views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
1
vote
0answers
28 views

Matrix: Area of a Triangle, which point to choose for cross multiplication

When given 3 points(vertices), which one should you pick to do the your calculations with. E.g.: $P1=(1,-1,1) P2=(2,1,-1) P3=(1,-2,-1) $ I can pick P1 -> P2 and P1 -> P3. Then do my cross ...
0
votes
1answer
132 views

Finding 2 vectors orthogonal to each other and a given vector

given vector: $u = <1, -7, 2>$ Find a vector $v$ and $w$ which are are orthogonal to each other and to $u$ I tried the following: $$ v \cdot u = <1,1,c> \cdot <1,-7,2> \rightarrow ...
0
votes
2answers
168 views

Proof in a Scalar Triple Product [closed]

For any three vectors $\vec a,\ \vec b,\ \vec c$, show that : $$[\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a]=[\vec a,\ \vec b,\ \vec c]^2$$ where $[\vec a,\vec b,\vec c]=\vec ...
5
votes
0answers
42 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
0
votes
1answer
33 views

Given 3 cross products of 3 vectors, how do you solve an expression of this format?

If you're given: $$a \times b = (2, -4, 2),\quad a \times c = (7, 13, -11),\quad b \times c = (1, 7, 1)$$ what properties of cross products or formulas can you use to solve $(2b - c) \times (3a + ...
0
votes
0answers
34 views

Cross-Product Intuition

Can anyone provide me with a little geometric insight as to why the ratio of the vector components of some vector perpendicular to two other should simply be the projections of those two vectors onto ...
0
votes
1answer
56 views

Projection on a hyperplan and a hypercube intersection

I need to project an array y onto a hyperspace defined by (a.x) = c where a is an array in R^N However, x needs to belong in the hypercube {0 <= x_i <= 1, for all i from 1 to n} Therefore from ...
0
votes
3answers
73 views

Cross product of vector functions

I was trying to make sense of a problem when I stumbled upon this on yahoo answers. I was just wondering if it was correct. If it is, can you please maybe explain why? ${\bf r}'(t) = \langle -5 \cos ...
2
votes
1answer
55 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
2
votes
1answer
115 views

Showing that a set of points equidistant to two other points form a plane.

Question: if p and q are two distinct points in space, show that the set of points equidistant from p and q form a plane. Work Done: Note: I'm pretty sure this can be done with vectors and cross ...
1
vote
3answers
410 views

How come the cross product of two planes is collinear with the direction vector of the line?

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.
3
votes
1answer
99 views

Show that $e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$

$$e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$$ I have to prove the above formula and am not sure where to start, may someone please help me! The full question is ...
0
votes
2answers
59 views

Show that angular momentum can be divided into center of mass and internal coordinate

This might belong in physics but I want to be sure I am approaching the math right. Given: show angular momentum can be divided into separate parts of the center of mass and internal coordinates. ...
1
vote
3answers
89 views

How do I plot a bunch of vectors in Maple and find the difference and cross product between them?

Title says it all. This is the assignment I'm trying to do. http://math.rutgers.edu/~shtelen/Teaching/Fall-2013/L1_instr.pdf Data I need to plot ...
1
vote
1answer
135 views

on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$

I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct. First, when I actually multiplied out $\vec{B} (\vec{A} ...
2
votes
3answers
445 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
53 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
244 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
126 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
39 views

Help clearing doubt about expansion of $\vec i\times(\vec a\times \vec i)$

I have this doubt in vector analysis I need help with. I know that cross product of a vector with itself is a null vector ($\vec a\times \vec a=\vec 0)$ as both point the same direction. Now consider ...
0
votes
1answer
1k 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
62 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
70 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
69 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
51 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 ...
8
votes
5answers
3k 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
167 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
138 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
98 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
430 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
93 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
69 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
45 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
173 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
449 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
89 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
167 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
66 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
97 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
98 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
149 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
163 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
7k 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
157 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 ...