For questions on cross products.

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How advanced should mathematical statements be for undergraduate research opportunities in mathematics?

I'm applying for a few undergraduate research opportunites in mathematics this Spring, and part of the application is discussing "the most interesting mathematical result you know of." All of the ...
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2answers
24 views

How to find an orthogonal vector C in $C^3$ relative to two other (given) vectors?

$A = [2,1,-i]$ $B = [i, -1, 2i]$ I need to find a C that is orthogonal to A and B. I've tried taking AxB, but this does not work. I get the vector C = (i, 1-4i, -2-i). The problem is that ...
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1answer
84 views

Is there a deeper meaning behind the “determinant” formula for the cross product?

We all know that for all vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R^3}$, if $(a_x,a_y,a_z)^\top$ is the component form of $\mathbf{a}$ and similarly $(b_x, b_y, b_z)^\top$ is the component form of ...
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2answers
44 views

Exterior Product vs Cross Product

I was confused about the relationship between a set of basis vectors in 3D, $ \left\{\hat e_1, \hat e_2, \hat e_3 \right\} $ and their exterior products. In my head, it makes sense that the identity ...
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4answers
25 views

Why we can conclude immediately that $x \in B$, if $(x, y) \in A \times B = B \times A$

The following statements are part of a proof involving cartesian products, specifically involving this theorem: $A \times B = B \times A \iff$ either $A = \emptyset$, $B= \emptyset$, or $A = B$ ...
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2answers
78 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
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0answers
54 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 ...
7
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1answer
90 views

Can the cross product be a matrix?

On the Wikipedia article on the cross product is says that a vector $a$ which is itself a cross product (that is $a=c\times d$), can be represented in the expression $a \times b$ for some other vector ...
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1answer
27 views

Integrating a cross product.

I am given that $\mathbf{k} . m \mathbf{q} \wedge \mathbf{\ddot{q}} = 0$ and my book says that integrating this wrt time gives $\mathbf{k}.m \mathbf{q} \wedge \mathbf{\dot{q}} = $constant. I don't ...
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2answers
53 views

How do you keep track of what vectors nabla ($\nabla$) should be working in on?

Take the following example: $$\vec\nabla\times(\vec A \times \vec B)$$ I assumed that this worked out to: $$\vec A(\vec\nabla.\vec B) - \vec B(\vec\nabla.\vec A)$$ Where, in both terms, Nabla ...
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1answer
17 views

Cross product problem

someone could show me the error in the cross products? For $U=x\hat{i}+y\hat{j}+z\hat{k}$, $V=x'\hat{i}+y'\hat{j}+z'\hat{k}$ and $((.))$=modulus, we have $$U \times V=((U))((V))sin(U,V).n = ...
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2answers
33 views

Bilinear form and cross product in hyperbolic geometry

I'm reading Patrick J. Ryan's Euclidean and non-Euclidean geometry, page 152. There is a bilinear form defined by $b\left( {x,y} \right) = {x_1}{y_1} + {x_2}{y_2} - {x_3}{y_3}$ on ${\mathbb{R}^3}$ and ...
0
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1answer
39 views

Cross product of two vectors, given magnitudes and angle

Problem Two vectors $\mathopen|{\overrightarrow{a}|=5.39} \ and \ \mathopen|{\overrightarrow{b}|=4.65} $ intersect and make a 120° angle. Find $\mathopen|{\overrightarrow{a}}\times ...
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0answers
7 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
22 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 ...
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0answers
13 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
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2answers
30 views

Vector equation with cross product and unit vector

Does anybody know how to solve the equation $\mathbf{a} + \mathbf{b} \times \hat{\mathbf{v}} = c \hat{\mathbf{v}},$ where $\mathbf{a}$ and $\mathbf{b}$ are given real vectors, for the unit vector ...
0
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1answer
50 views

In this situation, based on order of operations, would cross product happen first or dot product?

I got from wikipedia that the dot product is also referred to as the "scalar product" and that the cross product is also referred to as the "vector product". Can anyone confirm my inference on the ...
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0answers
27 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}) = ...
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0answers
54 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
1
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1answer
35 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...
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2answers
61 views

Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
2
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1answer
66 views

Cross Product in $\mathbb{R}^n$

I know that the cross product only exists in $\mathbb{R}^3$ and $\mathbb{R}^7$ but I am wondering what the actual definition of cross product is. That is, a cross product would be a function ...
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1answer
16 views

Find a plane defined by a point, a ray, and a vector starting from the point and parallel to another plane

I am trying to figure this out for implementation into a Graphics manipulator I've been trying to work out. I need to find a plane (a normal vector to the plane will suffice) and I know some of its ...
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2answers
87 views

Question about cross product and tensor notation

I am a bit rusty on tensor algebra and calculus and may use some wrong terminology, but I know that the cross-product can be expressed in tensor notation with the aid of the ...
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2answers
87 views

The formula for the magnitude of cross product, $\| u\times v \| = \| u \| \|v \| \sin \theta $ [closed]

Can someone show me a proof of the magnitude (length) of the cross product: $$\|u \times v \| = \| u \| \|v \| \sin \theta $$
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1answer
39 views

Cross product simplification

If you have two vectors, $A$ and $B$, then we can write the cross product as $A \times B$. Simplify the following expressions: $A \times (A \times (A \times B))$ $A \times (A \times (A \times (A ...
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0answers
27 views

Calculate a percentage based on intervals

I'm trying to calculate the relevance of an object based on it's location and a user's desired max distance. Also, I've relaxed the DB constraints, and retrieved from DB, objects that are twice as far ...
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1answer
29 views

Solving for first term in vector product

I'm trying to solve a system of equations for a physics application I've been working on, and I'm down to one thing left that's stumping me. Essentially, I need to solve $$A \times B = X$$ where $A, ...
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2answers
78 views

Proving a cross product satisfies the vector equation

Let vectors $u,v,w \in R^3$ Prove that $u \times (v \times w)$ must be a vector that satisfies the vector equation $x=sv+tw$ where $s,t \in R$ I have no idea where to go with this one, any tips?
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0answers
31 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 ...
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0answers
32 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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1answer
30 views

Show that Two Vectors Making Supplementary Angles?

I just need a start. I am not looking for whole prove, but it'd be more appreciated if I get one. Q. Use Theorem u . v = |u| |v| cos a and the trigonometric identity, cos (180-a) = -cos a, to ...
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0answers
17 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-Schwartz. Then we can define the angle, $\theta$, between non-zero vectors by requiring $a\cdot ...
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1answer
31 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
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1answer
36 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
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3answers
60 views

Where does the right hand rule appear in the “tensor” definition of the cross product?

When doing the cross product of two vectors according to the usual geometric definition of $\mathbf{A}\times\mathbf{B}$ being perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, it's pretty clear ...
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1answer
52 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, ...
0
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1answer
61 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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1answer
58 views

Vector cross product identity for $(a\times b)\cdot(c \times d)$

Prove that $(a\times b)\cdot(c \times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)$ I would appreciate some hints on how to solve this.I assume there is a method which does involve equating LHS ...
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600 views

Sphere equation given 4 points

Find the equation of the Sphere give the 4 points (3,2,1), (1,-2,-3), (2,1,3) and (-1,1,2). The *failed* solution I tried is kinda straigh forward: We need to find the center of the sphere. Having ...
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1answer
47 views

Is there any associative algebra that has all the algebraic properties of cross product?

Cross product is not associative, but is there any non-trivial associative algebra that has all the algebraic properties of cross product except the condition that would violate associativity and ...
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1answer
83 views

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 ...
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1answer
25 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
18 views

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
21 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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44 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 ...
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1answer
210 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
66 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 ...
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1answer
48 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 ...