For questions on cross products.

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3
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2answers
32 views

Sum of three cross products is zero.

Let $u,v,w\in \mathbb R^3$. Prove $u \times( v \times w)+v \times( w \times u)+w \times( u \times v) =0$ I guess things would work out if I just expanded as a ton of products. Is there a better way?
0
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0answers
21 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 $u \times w=u \times w$, 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 ...
1
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2answers
25 views

Use the cross product to find a parallel vector

I'm confused by this exercise here : Using the cross product, for which value(s) of t the vectors w(1,t,-2) and r(-3,1,6) will be parallel I know that if I use the cross product of two vectors, I ...
0
votes
1answer
11 views

A plane and the matter of vector crossing order

I have three 3D points $A$, $B$ and $C$ which are defining a plane. If I want to get the equation of the plane, firstly I need its normal vector. Is it matter if I do it with $AB \times AC$ or $AC ...
0
votes
1answer
21 views

Why is the cross product a x b dependent on size of vector a?

If I consider a x b = c, as a system where the vector b is rotating about an axis defined by vector a, and vector c shows the linear direction which vector b moves as it is rotating. The faster the ...
0
votes
1answer
13 views

Can you relate the cross product of two vectors to rotational motion.

Say you have a cross product a x b = c. Can you intepret this as the vector b spinning about the vector a. If the angle between them is close to zero then vector b is close to zero and is spinning ...
4
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0answers
42 views

What is the intuition behind the cross product?

I've researched lots of places and still cannot wrap my head around the cross product. The closest thing I have to an understanding of the cross product is that its a measure of how orthogonal two ...
1
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1answer
52 views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
0
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1answer
17 views

How to plot a contour in the plane defined by a vector cross-product

I have some contours of a structure in 3D that I would like to plot in 2D, but how do I get from 3D to 2D? In other words, I would like to plot the contour in the plane defined by the cross-product of ...
0
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0answers
8 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 ...
2
votes
4answers
34 views

Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
0
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1answer
67 views

Intersection of Three Planes proof

I'm supposed to be making a study guide answer for this question, but I'm struggling with proof. Show that the three planes intersect at the point provided that Note that the ...
2
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2answers
38 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
22 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
46 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
61 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
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1answer
32 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
294 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
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2answers
52 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
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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
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1answer
36 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
49 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
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3answers
28 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
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1answer
37 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
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1answer
24 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
57 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
33 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
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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 ?
1
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1answer
64 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
34 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 ...
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0answers
15 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
22 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
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1answer
65 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
94 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
51 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
39 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
79 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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0answers
26 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
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0answers
35 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
48 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
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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
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2answers
55 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
34 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
107 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 ...