For questions on cross products.

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5answers
25 views

Making sense of a cross product of three vectors

Because of the cross product of two vectors being another vector I can calculate $\vec a\times(\vec b\times\vec c)$ as well as $(\vec a\times\vec b)\times\vec c$. I know that the cross product is not ...
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2answers
30 views

Using Gram-Schmidt to compute the cross product of $3$ vectors in $\Bbb R^4$

I want to ask about vector multiplication (cross product) in $4$-d. I heard that Gram-Schmidt process is involved but I am not sure how the process is involved. The multiplication involves $3$ ...
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1answer
45 views

What is the 2d equivalent of vector multiplication? [on hold]

If two three-dimensional vectors, v1 and v2, are multiplied (i.e. dot product), the result will be a 3x3 matrix. If, instead, there are two three-by-three matricies, what is the corresponding ...
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3answers
28 views

cross product of vector and direction

We know that cross product gives a vector that is orthogonal to other two vectors. Let this vector denoted by $$|\vec{v} \times \vec{u}| = \vec{n}$$ Then $$\vec{n}\cdot \vec{u} = 0 $$ Everything okay ...
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2answers
137 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 ...
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1answer
30 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
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1answer
44 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
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6answers
6k 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 $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
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1answer
27 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 ...
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3answers
23 views

The magnitude of a triple product of two vectors

So I was going through a past exam for electrodynamics and a question for radiation came up and within it was the following magnitude of a triple product $ \lvert \hat{r} \times [\hat{r} \times ...
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2answers
31 views

Find Orthogonal Vector's Peak Point

I am given a 3-component vector $\vec v$. There are obviously an infinite number of orthogonal vectors to $\vec v$. I need to find the specific orthogonal vector, lets call it $\vec{x}$, in the plane ...
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1answer
33 views

Question about Lie bracket and cross product

Let $\chi(\mathbb{R^3})$ denote the vector space of all smooth vector fields on $\mathbb{R^3}$, and let $A$ be the subspace of $\chi(\mathbb{R^3})$ spanned by $\{X,Y,Z \}$ where \begin{align*} X ...
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1answer
19 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 ...
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0answers
23 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, ...
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3answers
44 views

Unordered cartesian product?

I have a set $\Omega=\{1;2;6\}$ and I want to define another set $A$ consisting of all triples $(a,b,c)$ with $a,b,c\in\Omega$, which contain exactly two 6's. My first attempt looked like this: ...
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1answer
32 views

Vector cross products proof

Explain why $U \times ( V \times W )$ must be a vector that satisfies the equation $X = sV + tW$, where $U$, $V$, and $W$ are vectors in $\mathbb R^3$.
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1answer
43 views

Multiplying Fractions Help

How could I find each product or quotient for this problem? I got $$\frac{3q}{2r^2}$$ as an answer The question is $$\frac{(4r)^2}{q} \times \frac{(3q)^2}{(8r)^4}$$ . So is my answer correct ?
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1answer
56 views

Cross product between a vector and a 2nd order tensor

I have been searching for quite a long time, and haven't been able to find any good reference about the cross product between a vector and a tensor: $$ \vec{a} \times \underline{T}= ...
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1answer
24 views

Intersection of Vectors

The vector $A=3 i + j - k$ is normal to the plane $M_1$, and the vector $B=2i - j + k$ is normal to a second plane $M_2$. Do the two planes necessarily intersect if they are both extended ...
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1answer
62 views

Geometric proof of the Cross Product magnitude

Most proofs of the magnitude of the cross product are algebraic in nature, I find I learn best visually / geometrically. Is there a breakdown of the proof of the magnitude of the cross product using ...
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1answer
29 views

Some algebraic properties of curl

Given vector fields $\mathbf E$ and $\mathbf H$ with curl $\mathbf E= - \frac 1 c \frac {\partial \mathbf H} {\partial t}$ and curl $\mathbf H= \frac 1 c \frac {\partial \mathbf E} {\partial t}$ ...
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0answers
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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 ...
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1answer
19 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} ...
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3answers
61 views

Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why ...
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4answers
47 views

$\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$

I have $\vec{a}\times(\vec{a}\times\vec{R})-\vec{b}\times(\vec{b}\times\vec{R})$, my textbook says that this equals $((\vec{a}\times\vec{a})-(\vec{b}\times\vec{b}))\times\vec{R}=-(a^2-b^2)\vec{R}$. I ...
0
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2answers
38 views

rotation matrix and vector - understand step calculation

I have an extremely equation, but I just don't understand which step they made to get to the last line. ${\bf W}$ and ${\bf V}$ are all 3d vectors. A is a rotation matrix. How did they get that ...
2
votes
1answer
63 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 ...
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2answers
70 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 ...
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0answers
23 views

How to find the surface area of a parameterized surface

Find the surface area of the following hyperboloid parameterized by $$\begin{align}x&=(\cosh{t})(\cos{\theta}) \\ y&=(\cosh{t})(\sin{\theta}) \\ z&=\sinh{t} \end{align}$$ ...
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0answers
22 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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5answers
15k views

What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
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1answer
44 views

Why should there be a 7-dimensional cross product in the context of exterior algebra?

The three-dimensional cross product can be viewed as the wedge product corresponding to the exterior power $\Lambda^2(\mathbb R^3)$. An explanation that I have come up with for the scarcity of cross ...
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1answer
48 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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3answers
47 views

Why is $a\times b = 0$ when $b = 2*a$?

In vector calculus, why is $a\times b = 0$ when you know that $b=2*a$? So how how do you know that a crossproduct of a vector and two times that vector is always zero?
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1answer
38 views

Shortest distance proof

Show that the shortest distance from a point $P$ to the line through $P_0$ with direction vector $\overrightarrow{d}$ is $$\frac{|\overrightarrow{P_0P}\times ...
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3answers
369 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 ...
0
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2answers
51 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
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1answer
24 views

Cross Product and Scalar Product proof

Hi i'm trying to prove the following equality where P,Q and R are any 3D vectors: PxQxR = (P.R)Q - (Q.R)P I find it easier by proving that the x coordinate of the left side is equal do the right ...
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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, ...
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2answers
101 views

Proof of identity: cross product of three vectors

A book I'm reading contains the following (paraphrased) \begin{equation} (a \times b) \times c = (a \cdot c)b - (b \cdot c)a \end{equation} This is supposed to follow from: \begin{equation} (a \times ...
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2answers
29 views

Manipulation of three vectors in $\Bbb R^3$

Please excuse the non-specific title, this is a rather long problem. So on our last exam in multivariable calculus, our professor gave us a very lengthy vector manipulation problem as a bonus. Seeing ...
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3answers
48 views

Intuitively, what makes two vector parallel

I have heard explanations such as it is when the cross product equals zero or that it is when one is a scalars multiple of the other but I have not seen an intuitive explanation. Is it when two ...
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8answers
342 views

Why is the cross product of two vectors orthogonal?

What is the intuition behind the fact that the cross product of two vectors is orthogonal? Every video I've seen just says it is orthogonal but they do not explain why. Since I have terrible memory, I ...
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0answers
29 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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1answer
65 views

Cross product for vector angular position?

The angular velocity of a particle $\omega = r \times v$ is a pseudovector because it is formed by the cross product of two vectors (position and linear velocity). Likewise the angular acceleration ...
2
votes
3answers
12k 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 $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that ...
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2answers
2k views

Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
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0answers
40 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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1answer
40 views

Cross product in uneven matrices

I don't need help with dot product, only the cross product section. Even a hint as to where to start would be great.
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0answers
32 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 ...