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Questions tagged [cross-product]

In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.

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16 votes
3 answers
105k views

Volume of tetrahedron using cross and dot product

Consider the tetrahedron in the image: Prove that the volume of the tetrahedron is given by $\frac16 |a \times b \cdot c|$. I know volume of the tetrahedron is equal to the base area times height, ...
0 votes
0 answers
24 views

Skew Symmetric Matrix vs. Cross Product

This might be more of a programming question in truth (at least, I suspect the answer is related to computer programming), but I figured I'd ask here. Why would someone choose to represent a cross ...
0 votes
1 answer
42 views

Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
1 vote
0 answers
51 views

Understanding the Relationship Between Cross Product Components and Differential Forms on a Membrane

I am struggling with a differential geometry and vector calculus concept involving the relationship between the cross-product components and differential forms. The specific context is a response ...
3 votes
0 answers
83 views

Identity for a scalar quintuple product?

I find myself needing to cross two pairs of vectors, and cross that result (so a normal of normals) and check whether each of two of the original points are on different sides of the plane it defines: ...
0 votes
0 answers
22 views

Surface area element in 4-dimensions

In Dirac's "General Theory of Relativity" (p. 40) he says "If we take two small contravariant vectors $\xi^\mu$ and $\zeta^\mu$, the element of surface area that they subtend is ...
23 votes
4 answers
6k views

Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
0 votes
2 answers
41 views

Loop integral of $\mathbf{r}\times d\mathbf{r}$ is equal to twice the enclosed area?

I came across the following integral in the literature for a loop $l$ that goes around a surface of area $A$: $$ \oint_l \mathbf{r}\times d\mathbf{r} = 2 A \mathbf{\hat{n}} $$ where $\mathbf{\hat{n}}$ ...
0 votes
0 answers
34 views

Tangent Vector, Principal normal vector, Binormal vector, and Torsion

So I'm trying to fully grasp how all these relate. My current understanding is that the tangent vector describes the direction in which the curve is going/curving. Meanwhile, the principal norm is ...
0 votes
0 answers
17 views

Cross product properties.

I just want to know if it's correct: let $\vec{v}=(\vec{a}+\alpha\vec{b})\times(2\vec{a}+\vec{b})$, with $\alpha\in\mathbb{R}$. If $||\vec{a}||=\sqrt{2},||\vec{b}||=1$ and the angle between $\vec{a}$ ...
0 votes
0 answers
32 views

What is a space which is the cross product $D^2\times S^1$?

While taking cross product of two one-spheres, $S^1\times S^1$ seems esasy to imagine/identify with a torus $T^2$, I struggle to make a picture of a $D^2\times S^1$, what kind of a space is that in ...
1 vote
2 answers
2k 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 $\...
3 votes
2 answers
1k 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 ...
0 votes
3 answers
36 views

Finding vector equation of a line

Show that the equation of a straight line passing through the point with position vector $\vec{b}$ and perpendicular to the line $\vec{r}=\vec{a}+\mu \vec{c}$ is of the form $\vec{r}=\vec{b}+\beta \...
0 votes
1 answer
1k views

Cross Product of two perpendicular vectors

Say I have two perpendicular vectors $\bf a$ and $\bf b$, and any vector $\bf c$, can anything be said about $(\bf a \times \bf b) \dot \bf c$?
14 votes
7 answers
12k views

Connection between cross product and determinant

When I calculate a cross product of two vectors in Cartesian coordinates, I calculate something that seems like the determinant of a 2x2 matrix. Is there any connection between the determinant and the ...
3 votes
4 answers
4k views

How to determine if a ray intersects a line?

I'm trying to determine how you could check if a ray, given an origin and a direction, and a line, given 2 points (Not a line segment) intersect. Ray: Origin(x,y), direction(x,y) Line: point1(x,y), ...
1 vote
2 answers
152 views

Question in pseudovectors

I learnt that under parity transformation a vector $\vec{A}$ <---(Parity)------> $-\vec{A}$ and a pseudovector can be written as $\vec{c}=\vec{A} $ $\times$ $\vec{B}$ and since A goes negative A ...
0 votes
0 answers
59 views

Prove that the cross product is a bilinear form

I want to show that the application $$\phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \quad \text{given by} \quad \phi(u, v)=u \times v$$ It is bilinear In first place, note that \begin{align*} ...
0 votes
1 answer
60 views

Interior, cross and outer products between two multivectors?

For two arbitrary multivectors $\mathbf u$ and $\mathbf v$, what are the definitions of the interior (or scalar) product $\mathbf u\cdot \mathbf v$, the cross product $\mathbf u\times \mathbf v$ (if ...
0 votes
0 answers
35 views

Reasons of computing smallest eigenvalue $R^TR$ instead of singular value

I have problems in understanding why author of this article uses smallest eigenvalue of a cross product matrix instead of a data matrix. I know that $SVD(AA^T)=UD^2U^T$, but I don't know why not ...
0 votes
2 answers
99 views

How do you show that $u+w+v = 0$ given $u \times v = v \times w = w \times u$ [duplicate]

Given that $u+w+v = 0$, I was able to prove that $u \times v = v \times w = w \times u$ by using the anti-commutative property. But I'm struggling a lot with how to approach to prove the converse. ...
0 votes
0 answers
36 views

Does cross product depend on the orthogonality of the basis vector [duplicate]

When I learn cross product, I find myself always using orthogonal basis vectors (e.g. $\hat{i}$, $\hat{j}$ and $\hat{k}$). But I am wondering does cross product depend on the orthogonality of the ...
4 votes
4 answers
854 views

Formula for cross product

The formula for the cross product of two vectors in $R^3$, $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$ is $$\det\begin{pmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\\ a_1 & ...
2 votes
1 answer
439 views

Formula for linear transformation of cross product

Motivated by this question and by my own calculations where I try to find what is the curvature of a rigid transformation to my curve I get to the following equation ($\alpha$ is a curve, $\phi$ a ...
1 vote
3 answers
348 views

How to see from definition that determinant is just volume?

The determinant can be defined recursively by Laplace expansion (LE), or equivalently, directly defined by Lebniz formula (LF) using parity of permutation. My question is: How do we see that these ...
1 vote
0 answers
77 views

How to algebraically derive the determinant expression for the cross product [duplicate]

I was reading this post: Deriving formula for cross-product. and I understand the derivation there of the formula and it's easy enough to just manually check that the resulting formula is the same as ...
2 votes
1 answer
82 views

Generalize an identity of cross products to $n$ dimensions

For any vectors $x,y,a\in\Bbb R^3$, let \begin{align}f(x,y)&:=x⋅y\\ f(x,y,a)&:=f(x,y)f(a,a) - f(x,a)f(y,a) \end{align} then $$\tag1 f(x,y,a)=(x×a)⋅(y×a)$$ To generalize this to 4D, I found by ...
33 votes
6 answers
67k views

What's the opposite of a cross product?

For example, $a \times b = c$ If you only know $a$ and $c$, what method can you use to find $b$?
1 vote
3 answers
102 views

Evaluation of surface integral $\iint F ds$; $F(x,y,z)=(x,y,z)$

Let $F(x,y,z)=(x,y,z)$. $\\[10pt]$ Evaluate: $\iint_{S} F ds$; where $S$ is the upper hemisphere of radius $3$, centered at the origin. I defined $\phi(u,v)=(\sqrt{9-v^2}\cos(u),\sqrt{9-v^2}\sin(u),...
0 votes
1 answer
89 views

Computing $U \times V$

Problem: Given $$ U = (2,1,3) $$ $$ U = (4,-2,2) $$ find $U \times V$. Answer: \begin{align*} U \times V &= \begin{vmatrix} i & j & k \\ 2 & 1 & 3 \\ 4 & -2 & 2 \end{...
7 votes
2 answers
9k 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 ...
2 votes
1 answer
2k views

Special Case of Lie-algebra

Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have ...
1 vote
0 answers
128 views

Cross-product Exercise

I am doing some problems in Spivak's calculus on manifolds, and I was stuck on this one. It says that: If $w_1, ... ,w_{n-1} \in \mathbb{R}^n$, show that $$| w_1 \times ... \times w_{n-1}| = \sqrt{\...
0 votes
2 answers
57 views

Determining value of unit vector that satisfies cross product equation

Consider the equation given by: $\langle 0,-1, 0 \rangle$ = $-qV \times \langle -1,0,0 \rangle$, where $\times$ denotes the cross product. I have to find a suitable standard basis vector $V$ that ...
0 votes
1 answer
118 views

Intuition for why a 90 degree rotation of a vector about an arbitrary axis can be expressed as 3 90 degree rotations of the vector's projections.

Given a unit vector $\hat{u}$ and a vector $\vec{v}$ perpendicular to $\hat{u}$, we can rotate $\vec{v}$ by 90 degrees around $\hat{u}$ with the cross product $\hat{u} \times \vec{v}$. Since the cross ...
1 vote
1 answer
34 views

Prove that the trajectory of $P$ is a circle in $3D$ and find its properties

A point $P=(x,y,z)$ starts off at $P(0)= (x_0, y_0, z_0)$. Its time derivative is given by $ \dfrac{d P}{dt} = a \times P $ where $a \in \mathbb{R}^3$ a unit vector, and $\times$ is the cross product....
0 votes
1 answer
70 views

On vector multiplication [closed]

In this video (check it out, it's worth it), F. Holmér nicely derives the dot and cross product (with some insights into quaternions, the wedge product and much more), just by using ordinary ...
1 vote
0 answers
29 views

A question about proof of bac-cab rule

I am trying to prove BAC-CAB rule. However, I am not sure about how can I guarantee that $\gamma=1$ is a solution for each vectors? If my proof is not correct, how should I proceed? I had the ...
16 votes
2 answers
697 views

How to prove that the cross product doesn't satisfy any kind of generalized associativity?

It's well known that the cross product in $\mathbb{R}^3$ doesn't obey the associative law of $$ A \times (B \times C) = (A \times B) \times C $$ We can define a "Generalized Associative Law" ...
0 votes
0 answers
243 views

Levi-Civita symbol for cross product generalisation, geometric interpretation, orthogonal subspace

Cross Product in $\mathbb{R}^3$ In $\mathbb{R}^3$ one can compute the coordinates of the cross product between $b_2$ and $b_3$ using the Levi-Civita symbol as follows: $$b_{1,i_1} = \sum_{i_2i_3} \...
0 votes
3 answers
69 views

Is it true that $\vec{\nabla}\times[(\vec{a}\cdot\vec{\nabla})\vec{a}]=\vec{\nabla}\times[\vec{a}\times(\vec{\nabla}\times\vec{a})]$?

I'm reading some notes on particle physics by a university professor, and after doing calculations I've reached the conclusion that in order for one of his claims to be true, this equality would have ...
0 votes
0 answers
86 views

Prove a subset $E$ of $\mathbb{R}^3$ is exactly a plane if there are vectors $u,v,w\in\mathbb{R}^3$ such that $v$ and $w$ are linearly independent and

Prove that a subset $E$ of $\mathbb{R}^3$ is exactly a plane, if there exists vectors $u,v,w \in \mathbb{R}^3$, such that $v$ and $w$ are linearly independent and $$E = u + \mathbb{R}v + \mathbb{R}w$$ ...
4 votes
2 answers
504 views

What is the operation between a bivector and a vector that outputs another vector?

I'll start with a physical example. Let's say we have the angular velocity (in 3D euclidean space with orthonormal basis spanning it), which has the bivector representation $$\Omega = \omega_x \mathbf ...
0 votes
1 answer
1k views

Is taking sum inside cross product valid?

I have a sum of a cross product over one of the multipliers. In this case it has a physics application being a sum over magnetic moments, $\vec{\mu}$, to give magnetisation, $\vec{M} = \sum_i\vec{\mu}...
4 votes
1 answer
264 views

Show that the sum of these four vectors is $0$.

Four vectors are erected perpendicularly to the four faces of a general tetrahedron. Each vector is pointing outwards and has a length equal to the area of the face. Show that the sum of these four ...
0 votes
0 answers
57 views

Cartesian product of two balls

Consider the following exercise: Let $(a,b)\in\mathbb{R}^n\times\mathbb{R}^m$ be an arbitrary point and let $\varepsilon>0$. Show that there are positive real numbers $\varepsilon_1,\varepsilon_2$ ...
0 votes
1 answer
78 views

Explanation of norm of cross product formula

I have a line between two points given as $(x_1, y_1)$ and $(x_2, y_2)$ in Python code: ...
1 vote
0 answers
64 views

This function has a fixed point, $F^4(x)=x$. Why?

Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
0 votes
1 answer
32 views

Can you characterize a vector field $v$ with uniform curl as $v(x) = a + b \times x$?

Let $v:\mathbb R^3 \to \mathbb R^3$ a vector field such that $\forall x\in\mathbb R^3, (\nabla \times v) (x) = \rho~$ where $\rho$ is a constant. Example here. Can you characterize $v$ simply? What I ...

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