# 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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### 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, ...
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### 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 ...
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### 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
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 $\mathbf{\hat{n}}$ ...
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### 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 ...
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### 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 $\vec{a}$ ...
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### 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
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### 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 $(\bf a \times \bf b) \dot \bf c$?
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### 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 ...
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### 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
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### 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 ...
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### 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*} ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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
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### 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....
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### 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 ...
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### 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 ...
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### 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" ...