For questions on cross products.

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2
votes
2answers
30 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
0
votes
1answer
21 views

Understanding step in deriving the formula of the curvature.

Last formula on third page of the document: Computation of $\vec{r'}(t)\times \vec{r''}(t)$ From the previous two formulas and using the properties of cross products we see that ...
2
votes
1answer
44 views

Dot product of two vectors obtained by cross product

How can I prove the following identity: $$(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{c}\cdot\vec{a})(\vec{b}\cdot\vec{d})-(\vec{b}\cdot\vec{c})(\vec{a}\cdot\vec{d})$$
1
vote
3answers
49 views

Cross product spherical coordinates

I can't wrap my head around the result of the cross product of two vectors in spherical coordinates. Is it a vector or something that I can represent geometrically? For example, given two vectors in ...
0
votes
1answer
24 views

Scalar triple product of quaternion scalar parts

I'm reading this paper about quaternion and 3D rotation with unit quaternions, \begin{eqnarray*} && \dot{q} = (q, {\bf q}) \\ && \dot{r} = (r, {\bf r}) \\ && \dot{r'} = ...
3
votes
0answers
70 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
votes
3answers
286 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 ...
1
vote
2answers
49 views

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

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 $3$ ...
0
votes
3answers
32 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 ...
0
votes
1answer
34 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 ...
0
votes
1answer
48 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 ...
4
votes
1answer
28 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 ...
0
votes
3answers
26 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 ...
0
votes
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 ...
1
vote
1answer
35 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 ...
1
vote
1answer
21 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 ...
0
votes
0answers
25 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, ...
2
votes
3answers
52 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: ...
1
vote
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 $\mathbb R^3$.
1
vote
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 ?
1
vote
1answer
59 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}= ...
2
votes
1answer
31 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 $\mathbf H= \frac 1 c \frac {\partial \mathbf E} {\partial t}$ ...
0
votes
1answer
25 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 ...
1
vote
0answers
13 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
vote
1answer
20 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} ...
0
votes
1answer
63 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 ...
3
votes
3answers
67 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 ...
2
votes
4answers
49 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
votes
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
65 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 ...
1
vote
0answers
24 views

How to find the surface area of a parameterized surface

Find the surface area of the following hyperboloid parameterized by $$\begin{align}x&=(\cosh{t})(\cos{\theta}) \\ y&=(\cosh{t})(\sin{\theta}) \\ z&=\sinh{t} \end{align}$$ ...
0
votes
0answers
24 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 ...
1
vote
1answer
46 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 ...
0
votes
3answers
48 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?
0
votes
1answer
39 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 ...
0
votes
2answers
54 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 ...
0
votes
1answer
27 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 ...
6
votes
2answers
103 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 ...
1
vote
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 ...
2
votes
3answers
53 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 ...
0
votes
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
votes
1answer
71 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 ...
3
votes
8answers
376 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 ...
0
votes
1answer
42 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.
2
votes
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 ...
0
votes
1answer
46 views

Linear Algebra: Compute Area of Parallelogram

I have this one Linear Algebra question that is asking me to compute the area of a parallelogram defined by 4 vectors. Here is the question: Let $\vec{u}=\begin{bmatrix}a\\b\end{bmatrix}$ and ...
1
vote
2answers
47 views

vectors in $3$ dimensions such that $v+(v \times u)=u$

Let $i,j,k$ denote the usual three unit vectors in $\mathbb R^3.$ 1) Find all vectors $v \in \mathbb R^3$ such that $v+(v \times i)=j$. 2) Suppose vectors $v$ and $u$ belong to $\mathbb R^3$ and ...
1
vote
0answers
43 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} ...
1
vote
1answer
36 views

Problem with understanding a cross product related problem

I have a question that follows like this: Let $U = (0,1,2)$ and $V = (1,-1,-2)$. Assume that $V \times W = (3,1,1)$ Is there enough information provided in order to determine $(U \times V) ...
0
votes
2answers
36 views

Cross Product of two vectors

I have these two vectors $$\vec{a}_1=\frac{a}{2}\hat{x}+\frac{\sqrt{3}}{2}a\hat{y}$$ $$\vec{a}_2=c\hat{z}$$ I know $\vec{a}_1\times\vec{a}_2$ is equal to: ...