For questions on cross products.

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3answers
176 views

How can I prove that two vectors in $ℝ^3$ are linearly independent iff their cross product is nonzero?

Here's my attempt: Let $𝒙 = (x_1, x_2, x_3)$ and $𝒚 = (y_1, y_2, y_3)$ The cross product of $𝒙, 𝒚$ is $𝒙⨯𝒚=(x_2y_3-x_3y_2, x_3y_1 - x_1y_3, x_1y_2 - x_2y_1)$ And linear independence of $𝒙, ...
2
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1answer
74 views

Generalising the cross product to infinite dimensions, does $v \times v = 0$ hold also in infinite dimensional spaces

Consider I have a vector space $V$ with inner product and a bilinear map $b : V \times V \to V$ i) such that if $z = b(u,v)$ for two $u,v \in V$, then $$ z \perp u \quad \mbox{ and } \quad z \perp ...
1
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1answer
28 views

Points defining plane - starting step? [closed]

If the points $P, Q, R$, not all lying on the same straight line, have position vectors $a, b, c$ respectively, show that $(a \times b) + (b \times c) + (c \times a)$ is a vector perpendicular to the ...
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2answers
68 views

Cross Product for Biot-Savart Derivation of Current Loop

Biot-Savart's law can be used to determine the magnetic field produced by a figure at a point. Introductory physics texts integrate $dB$ to obtain $B$ where $dB$ = $\frac{I\mu_{0}}{4\pi r^2} * dl ...
2
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2answers
41 views

Ratio of area formed by transformed and original sides of a parallelogram

I am interested in finding the ratio of area formed by transformed and original sides of a parallelogram, given by: $$\frac{\|Ma\times Mb\| }{\| a\times b \|}$$ $M$ is a $3 \times 3$ matrix and $ a, ...
2
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1answer
43 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
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1answer
21 views

Visualizing cross product of points in 3-Space

If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically? I'm having a little trouble visualizing this in ...
0
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2answers
28 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 ...
9
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2answers
145 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 ...
2
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2answers
70 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 ...
0
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4answers
27 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
85 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
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 ...
7
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1answer
152 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 ...
3
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1answer
47 views

Set-theoretic questions about the definitions of crossed-product $ C^{*} $-algebras and group $ C^{*} $-algebras.

In his book Crossed Products of $ C^{*} $-Algebras, Dana P. Williams defines the crossed product of a $ C^{*} $-algebra $ A $ by a locally compact group $ G $ as the completion of $ {C_{c}}(G,A) $ ...
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1answer
215 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 ...
0
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2answers
64 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 ...
0
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1answer
18 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 = ...
1
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2answers
56 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
64 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
11 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
23 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 ...
0
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0answers
26 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 ...
0
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2answers
32 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
567 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
30 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
96 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
2
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1answer
41 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 ...
3
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2answers
167 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
72 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 ...
0
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1answer
38 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
147 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 ...
0
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2answers
108 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
48 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 ...
0
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1answer
35 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, ...
0
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2answers
97 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
41 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
34 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
34 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
18 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 ...
0
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1answer
48 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} ...
0
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1answer
37 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!
0
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3answers
92 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 ...
2
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1answer
55 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
82 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
76 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 ...
6
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5answers
1k 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 ...
1
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1answer
49 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 ...
3
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1answer
103 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 ...
1
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1answer
31 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 ...