For questions on cross products.

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Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
1
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1answer
16 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...
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0answers
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Properties of cross product ${\rm i}(a\times a^*)$

Given a complex 3-vector $a\in\mathbb{C}^3$, let $b$ be the following vector $$b={\rm i}(a\times a^*)$$ where $a^*$ is the element-wise complex conjugate of $a$. As can be easily shown by ...
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0answers
17 views

Cross product query.

In cross product, we do it like: a vector x b vector = a*b*sin(theta). From where does this sin(theta) came from? Can someone please derive the cross product and explain it.
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0answers
32 views

covariance of cross product of two vectors

I have two independent vectors in 3D and know the covariance matrix of each. What will be the covariance of cross products of above vectors. In particular what will be the covariance of cross ...
1
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1answer
39 views

Product rule for gradient of cross product

The book I am reading gives a list of product rules, among them the following: $$\nabla \cdot (v\times w) =(\nabla \cdot v) w-v\nabla \cdot w.$$ However, the left-hand side is a number whereas the ...
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1answer
57 views

Programming constraints in video game. How are these two equations equal?

I'm currently working on programming a game that uses a physics engine (NAPE). Inside of that engine there are constraints that you can program. In order to program those you need a somewhat ...
0
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1answer
21 views

If $u$ and $v$ are vectors in $R^3$, simplify the expression $(u+v) \times (u-v)$ as much as possible.

Here is my thinking process for answering this question: Cross product is neither commutative nor associative. Hence I cannot do any algebraic operations on this expression. However I know that cross ...
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2answers
52 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
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3answers
38 views

Calculus - Components of a unit vector

Determine the components of a unit vector perpendicular to (0, 3, -5) and (2, 3, 1). I think I should be using either cross or dot product, but am unsure on what to do from there.
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2answers
64 views

Given two unit vectors, find a vector perpendicular with additional constraint

Given two unit length vectors find a perpendicular vector of unit length. I want to know if there's a way to do this without using a square root operation (avoid a normalization operation). Since the ...
3
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1answer
35 views

cross product in cylindrical coordinates

Hi i know this is a really really simple question but it has me confused. I want to calculate the cross product of two vectors $$ \vec a \times \vec r. $$ The vectors are given by $$ \vec a= a\hat ...
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1answer
32 views

Clarify Right hand Rule

I was just wondering whether in the right hand rule are all 3 vectors perpendicular to one another or is it simply one way, i.e. $A \times B=C$, would it be right to also say $C \times A=B$ and $C ...
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2answers
28 views

Writing a vector as the sum of two other vectors.

Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and ...
0
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1answer
30 views

Cross Product in 3D

Hi! I am currently working on some calc2 online homework problems concerning the cross product. I understand how the cross product works, but I am not sure how to apply it to this question. I know ...
2
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1answer
30 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
0
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1answer
46 views

Image and Kernel of different linear maps and their dimension

I'm trying to determine the image and the kernel of different linear maps. I understood well the theory but I can not transfer the knowledge of the books I have read to specific linear maps. 1) ...
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0answers
10 views

Relational Algebra Cross Join Definition. Need Clarification

I'm currently reading Lee Wilkinson's Grammar of Graphics, and I'm having trouble understanding his mathematical definition of the cross join. I understand the input and output, but the relational ...
2
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1answer
130 views

How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
1
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1answer
40 views

Why is the cross product contained in orthogonal complement?

Let $(V,\langle,\rangle)$ be the $\mathbb R^3$ with the standard bilinear-form and let $W \subset V$ be a two dimensional spanning set given by $v = (x_1,x_2,x_3)$ and $w = (y_1,y_2,y_3)$ and the ...
0
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1answer
38 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
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1answer
47 views

What is the difference between $|a|$ and $|| a ||$

So I am doing maths involving cross-product and dot-product and I came across the above two notation as in $||u\times v|| = ||u|| ||v|| \sin a$ and $u\cdot v = |u| |v| \cos a$. What is the difference ...
3
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1answer
177 views

Name of an identity for traceless matrices in $\mathbb{R}^3$?

While working on a more compact presentation of a derivation in the context of incompressible fluid flow we tried to simplify things by introducing intermediate steps instead of writing out lengthy ...
3
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0answers
19 views

Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
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2answers
41 views

About matrix $R$, what is this called: $R^TR$? What is it for?

I am doing singular value decomposition on a matrix $R$. The first step is to compute such a matrix $R^TR$. What is this matrix? A reference told me this is cross product of matrix R. I use a ...
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2answers
33 views

How to find a normal vector from an equation in the form f(x,y)?

If I have an equation $f(x,y)$ which given the $x$ and $y$ coordinate, it gives you the $z$ coordinate. How can I find the normal (directional) vector of the the point $(x,y,f(x,y))$? This would be ...
3
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1answer
34 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
0
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2answers
51 views

Adding two vectors such that the resulting vector is perpendicular to a third vector

Let $$a = (-3, 3, 1)$$ $$b = (1, 4, -4)$$ $$c = (2, 1, -3)$$ For which values of $t \in \Re$ is $b + tc$ perpendicular to a? For a vector to be perpendicular to $a$, the dot product of that ...
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2answers
105 views

Line integrals, cross products, surface integrals and Stoke's Theorem related problem?

The vector field $\vec{F}(\vec{R})$ is defined as being equal to the line integral over some simple closed curve $C$: $$\vec{F}(\vec{R})=\oint_C\|\vec{r}-\vec{R}\|^2d\vec{r}.$$ We show that there ...
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0answers
9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
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2answers
809 views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
1
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1answer
17 views

cross-products versus units of measure

If I draw 2 perpendicular line segments on the ground, 3 meters and 4 meters, how far into the sky does their cross-product extend? What if instead the line lengths are 300 cm and 400 cm? Can ...
0
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1answer
57 views

Prove that $g(t) \times \frac{d}{dt} g(t) = 0$

If the vectorial function $r = g(t)$, with values in $\mathbb R^3$ and where $t\in\mathbb R$, is a solution of the differential equation $\frac{d^2}{dt^2} r(t) = t^2 r(t)$, such that $g(0) = 0$, ...
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1answer
80 views

Can cross products be defined without coordinates?

I recently learned about cross products and understood that cross products can be computed without an origin and coordinates in three dimensions, like vectors can be defined without coordinates. But ...
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0answers
36 views

Given two lines, how do I find the plane?

$$r_1(t) = \langle t, 2t, 3t\rangle$$ $$r_2(t) = \langle3t, t, 8t\rangle$$ I found $\mathbf{n} = \langle13,1,-5\rangle$ Can I just plug in say $P_0 = (0,0,0)$ and get $13x+y-5z = 0$?
3
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1answer
51 views

Cross product and right hand rule

Is there a simple proof that the cross product (defined as the usual determinant) always obeys the right hand rule?
0
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1answer
48 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
4
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2answers
80 views

Cross product in > 3d

What exactly would we get by calculating the cross product of vectors in $R^n, n>3$ using the formula $\vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n$ $\vec n$ being a vector normal ...
0
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2answers
54 views

Vector triple product = 0

Let $U, V, W$ be three non-zero vectors, no two of which are parallel. Under what conditions is $U\times(V\times W) = 0$?
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1answer
50 views

How do you find the max value of a length of a vector?

I have a vector $v = 7j$ and a vector $u$ with a length of 5 that starts at the origin and rotates in the $xy$-plane. How am I supposed to find the max value of the length of the vector $|u \times ...
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0answers
102 views

What is does the transformation $[\mathbf{a}]_{\times}$ do?

https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication I'm curious about the matrix $[\mathbf{a}]_{\times} \stackrel{\rm def}{=} ...
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2answers
300 views

Equation of plane that goes for intersection of 2 planes and is perpindicular to another plane

Really don't know what to do here, went to a tutor neither did he. Okay first the problem: Find the equation of the plane that passes through the line of intersection of the planes x − z = 2 and y + ...
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0answers
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Given the normal vector n(s), determines the curvature k(s) and the torsion

Given the normal vector n(s) of a curve $\alpha$, with non zero torsion everywhere, determines the curvature k(s) and the torsion $\tau$(s) of $\alpha$. I am first trying to show the following which ...
2
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1answer
45 views

Relation between $\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ and $\left(\vec{a}\times\vec{b}\right)\times\vec{c}$

The operation $\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ can be simplified to $\vec{b}\left(\vec{a}\cdot\vec{c}\right) - \vec{c}\left(\vec{a}\cdot\vec{b}\right)$ and can easily be remembered by ...
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0answers
37 views

Cross Product Component Values

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And ...
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2answers
48 views

Prove that vectors x,y are linearly dependent exactly when …

Prove that vectors $\vec{x},\vec{y}$ (belonging to $\mathbb{R}^3$) are linearly dependent only if the following is true $$ \begin{vmatrix} x_1&y_1 \\ x_2&y_2 \end{vmatrix} ...
4
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0answers
133 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
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2answers
294 views

Finding the volume of a pyramid (the vector way)

The problem I have 4 points $ P \; (-1,2,0) \\ Q \; (2,1,3) \\ R \; (1,0,1) \\ S \; (3,-2,3) $ and I want to find the volume of a pyramid. What I'm most concerned here is the appropriate strategy ...
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1answer
121 views

Projective geometry. Interpretation of a cross product between a line coincident with a point

Let $p \in \mathcal{P}^2$ be a point in projective 2-space coincident with a line $l\in\mathcal{P}^2$ such that $l^\top p = 0$. What does $l \times p$ mean? For example, $p = ...
2
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2answers
150 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...