For questions on cross products.

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4
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1answer
29 views

Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$ \frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b), $$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
4
votes
1answer
100 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 ...
2
votes
1answer
114 views

Is there a nice meaning to the geometric triple product?

Using geometric algebra, I can easily find the geometric tripleproduct of three vectors $a,b,c \in \mathbb{R}^3$ to be $$abc = a \left(b \cdot c \right) - b \left( c \cdot a \right) + c \left( a ...
2
votes
1answer
59 views

how $3i \times 3i = 9i \times i$? (i is the unit vector and $\times$ is cross product)

$i$ is the unit vector; didn't know how to write it. I'm reading a text and somewhere it uses something like $ai \times bi = (ab)i \times i$ (implicitly). I can see why this is true geometrically, ...
2
votes
1answer
378 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 ...
1
vote
1answer
27 views

vector triple product does not go from $[\dot{M} \times [H \times M]]+[M \times [H \times \dot{M}]]$ to $2\dot{M}(H,M)$

I have the following term $ [M \times [H \times M]] $ under the time derivative. After using the rule of the derivate of the product I get: $$[\dot{M} \times [H \times M]]+[M \times [\dot{H} \times ...
1
vote
1answer
27 views

Surface integrals, what happens to the $\sin \theta$ part

In the derivation of the formula for surface integrals we find that the surface $S$ of a parametric function $f(u,v)$ for the area $D$ can be found using the following entity: $$A(S) = \iint_{D} ...
1
vote
1answer
76 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 ...
5
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0answers
101 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 ...
4
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0answers
257 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 ...
3
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0answers
73 views

Are all 7-dimensional cross products isomorphic?

Let $\times$ be this 7-dimensional cross product and let $\hspace{.04 in}f$ be a bilinear map on $\mathbb{R}^7$ which satisfies the orthogonality and magnitude conditions. Does there necessarily ...
3
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0answers
47 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 ...
2
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0answers
34 views

Is this tensor identity true?

If We have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ and a symmetric positive definite Matrix $\boldsymbol{M}$ I was wondering if the expression $((\boldsymbol{a}\times \boldsymbol{b}) \cdot ...
2
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0answers
223 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
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0answers
8 views

Discrete cross-correlation: How to do cross correlation in FFT with segmented windows?

I have a discrete signal: $x=[i_1, i_2, ..., i_n]$ I would like to do the cross correlation between sub-groups of (k number of) consecutive number of $x$, for example: $xcorr([i_1, i_2, ..., i_k], ...
1
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0answers
31 views

units in math, cross product

The cross product of two vectors has length equal to the area of the parallelogram they generate. The cross product is also a vector and thus has dimensions. But the units of those dimensions are ...
1
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0answers
15 views

Compute $f\nabla\times\mathbf{r}$

I am trying to compute $f\nabla\times\mathbf{r}$ for $$ f(x,y,z) = \left(x^2+y^2+z^2\right)^{-\frac{3}{2}} \\ \mathbf{r} = x\mathbf{i}+y\mathbf{j}+z\mathbf{k} $$ I have computed $\nabla ...
1
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0answers
64 views

How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
1
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0answers
34 views

How to find the surface area of a parameterized surface

Find the surface area of the following hyperboloid parameterized by $$\begin{align}x&=(\cosh{t})(\cos{\theta}) \\ y&=(\cosh{t})(\sin{\theta}) \\ z&=\sinh{t} \end{align}$$ ...
1
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0answers
48 views

Integration of multivariate Gaussians with cross terms

I'm stuck with the following integral: $I=\int ... \int exp\Big(-\frac{1}{2} \sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} x_{t}+\sum \limits_{t=1}^{n} x_{t}^T{\Sigma_{x}}^{-1} z_{t} -\frac{1}{2} ...
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0answers
61 views

Using the exchange lemma provides weird result

I know that if we have a square $n \times n$ matrix $A$ in F and two vectors $v,u$ in $\mathbb F^n$, following the exchange lemma we can show that: $$\langle Au,v\rangle = \langle u, \ A^*v\rangle ...
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0answers
17 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
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0answers
28 views

Simple indefinite integral of a vector function

I am having trouble with this simple integration. I am not sure of the process or steps to follow to solve this type of problem: If $\mathbf{V}(t)$ is a vector function of $t$, find the indefinite ...
1
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0answers
44 views

'Simple' Vector analysis and I need a little help prooving an identity.

I have to prove that $$\vec{S}(\vec{r}) = -\frac{c^2}{\omega} \bigg( u(\vec{r})\vec{\nabla}\phi(\vec{r}) + \frac{i}{2}\vec{\nabla}u(\vec{r}) \bigg) ~~~~(1)$$ given that $$\vec{S}(\vec{r}) = ...
1
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0answers
125 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
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0answers
32 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.
1
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0answers
123 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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0answers
39 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 $13x+y-5z = 0$?
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0answers
45 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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0answers
35 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
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0answers
24 views

Homology of $T^3$ generated by three copies of $T^2$

Why is the second homology of $T^3=S^1\times S^1\times S^1$ generated by $S^1\times S^1\times \{\mathrm{pt}.\}$, $S^1\times \{\mathrm{pt}.\}\times S^1$ and $ \{\mathrm{pt}.\}\times S^1\times S^1$? ...
0
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0answers
20 views

Decompose cyclic sum of crossproducts into two cyclic sums?

Suppose you have $6$ points $a_i\in\mathbb{R}^3$ $i\in\{1,..,6\}$ such that all triangles with vertices $0, a_i, a_{i+1}$ for $i\in\{1,..,5\}$ do not degenerate (I dont know if this assumption is ...
0
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0answers
23 views

Construct 3D plane from 2 points and minimize angle of two vectors with its normal

I have as input two points $P, Q \in E^3$ and two vectors $\vec{v}_1, \vec{v}_2 \in R^3$. I need to construct a plane $(\vec{n}, d)$ such that the two points are in the plane and the angles between ...
0
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0answers
26 views

Using the Jacobian matrix to find surface area without a change of basis.

http://mathinsight.org/parametrized_surface_area_examples In reading through the example in the above link, it's straightforward to find the surface area for a cone as follows. Find the surface ...
0
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0answers
20 views

Norm of cross product

We define cross product $a\times b$ as a determinant of the following matrix: $$\begin{pmatrix} i&j&k\\ a_1&a_2&a_3\\ b_1&b_2&b_3 \end{pmatrix} $$ where $i,j,k$ forms the ...
0
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0answers
32 views

Angle using cross product

I have a situation. Please refer to a figure below: I have r1, r2, Ɵ1, Ɵ2 as well the reference line. I want to find out angle Φ(phi). i.e. (angle PBA). Edit_1 The link provided solves the ...
0
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0answers
28 views

Problem with triple cross product proof

When trying to prove bac-cab rule, I get to point, where I don´t know, what is true. I have $$ (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m=a_jb_ic_j-a_lb_lc_i $$ but when $$ ...
0
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0answers
21 views

Proof of the reconcilation of the geometric form of cross product with the algebraic form.

In Arfken's "Mathematical Methods for physicists" he stated that: $(A\times B)\cdot(A\times B) = A^2B^2-(A\cdot B)^2=A^2B^2-A^2B^2\cos^2(\theta)$ How did he arrived to that? He said that he is ...
0
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0answers
33 views

Theorems for Perpendicular and Parallel Vectors

I know the following about lines: If two lines are parallel to a third line, then they are parallel to each other. If two lines are perpendicular to a third line, then they are parallel to each ...
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0answers
35 views

Uniqueness of cross product

My professor in E&M told us that the cross product is the only possible product of vectors that produces another vector. I didn't ask him what he meant by that because he is Russian and can't ...
0
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0answers
50 views

Vector, C, orthogonal to both vectors A and B (which are orthogonal to each other)

Vectors A and B are orthogonal. To then find a vector C which is orthogonal to both A and B do I use the triple product (A x B = C) or do I use the dot product again and set equal to zero?
0
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0answers
37 views

Properties of vectors in $\mathbb{R}^3$ related by cross-products

If $u, v$ and $w$ are vectors in $\mathbb{R}^3$ such that $u \times w=v \times w$, which of the following will be true: a) $\pi_w u= \pi_w v$ b) $\pi_u w= \pi_v w$, where $\pi_x$ is the projection ...
0
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0answers
21 views

cross product of material derivative

I am looking to evaluate $\vec{n} \times \dfrac{D\vec{u}}{Dt}$ where $\dfrac{D}{Dt}$ is the material derivative. Can I bring the cross product into the derivative and rewrite the expression as ...
0
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0answers
43 views

Given verticies find the area of the triangle formed

When I looked at this problem I didn't think it seemed all that hard until I actually tried it. The problem is this: Given the rectangular vertices $O(0, 0, 0), P(-1, 2, -3), Q(-2, 3, -4), R(0, 0, ...
0
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0answers
87 views

The gradient of the magnitude of the cross product of a constant vector and the position vector

If $ \underline c$ is a constant vector and $\underline r$ is the position vector. How can I show that $ \lvert \underline c \land \underline r \rvert grad \lvert \underline c \land \underline r ...
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0answers
30 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in ...
0
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0answers
18 views

I have problem in understanding the relation?

If we have a relation like this: $$\frac { \partial }{ \partial x_\beta } (\varepsilon_{ij\alpha} x_j T_{\alpha \beta}) =\left[ \nabla \cdot(\vec x \times \overset {\leftrightarrow}{T} ) \right]_i$$ ...
0
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0answers
45 views

Proof dealing with orthogonality and cross product

If $v$ is orthogonal to vectors $x$ and $y$, then $v$ is a scalar multiple of $x \times y$. So far I have that: $v\cdot x=v_1x_1+v_2x_2+v_3x_3=0$ and $v\cdot y=v_1y_1+v_2y_2+v_3y_3=0$ $x \times y ...
0
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0answers
38 views

The identify with cross product : how to understand?

The problem come from the notations in Hansbo's paper (page 197) and his another paper. Given a domain $\Omega$, the function $u\in H_0^1(\Omega)$. $\Omega$ is divided into 2 subdomains ...
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0answers
24 views

Proving formula for length of crossed product

Suppose we are working in $\mathbb{R}^3$ and have the defined the usual scalar product and proven Cauchy-Schwarz. Then we can define the angle, $\theta$, between non-zero vectors by requiring $a\cdot ...