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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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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 ...
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Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
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115 votes
7 answers
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Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal ...
VF1's user avatar
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40 votes
4 answers
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Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have ...
goblin GONE's user avatar
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23 votes
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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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25 votes
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Cross product in $\mathbb R^n$

I read that the cross product can't be generalized to $\mathbb R^n$. Then I found that in $n=7$ there is a Cross product: https://en.wikipedia.org/wiki/Seven-dimensional_cross_product Why is it not ...
Anna's user avatar
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34 votes
8 answers
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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 $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
jjepsuomi's user avatar
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Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ ...
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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 $b$?
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5 answers
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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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60 votes
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Why is cross product defined in the way that it is?

$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a ...
koe's user avatar
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52 votes
4 answers
58k views

Understanding Dot and Cross Product

What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?
David McGraw's user avatar
11 votes
3 answers
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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 ...
Tarek's user avatar
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37 votes
2 answers
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Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
William Chang's user avatar
15 votes
8 answers
14k views

Scalar triple product - why equivalent to determinant?

I'm looking at the scalar triple product and I'm wondering: is there any demonstration (possibly a simple one) that $$ \mathbf{a} \cdot \left(\mathbf{b} \times \mathbf{c} \right)= \begin{bmatrix} ...
John Smith's user avatar
43 votes
5 answers
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Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of two vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
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15 votes
6 answers
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Motivation for construction of cross-product (Quaternions?)

I'm trying to present a narrative that brings the (3D) Cross Product into existence. "Given two vectors $\mathbf u$, $\mathbf v$, how to construct a vector perpendicular to both?" ... looks like a ...
P i's user avatar
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15 votes
2 answers
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Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = \...
IremadzeArchil19910311's user avatar
13 votes
2 answers
1k views

Deriving BAC-CAB from differential forms

I've recently begun reading up on differential forms in a physics context, and my resources said that one can often derive vector identities from differential forms. For instance, $\nabla \cdot (\...
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Octonionic formula for the ternary eight-dimensional cross product

A cross product is a multilinear map $X(v_1,\cdots,v_r)$ on a $d$-dimensional oriented inner product space $V$ for which (i) $\langle X(v_1,\cdots,v_r),w\rangle$ is alternating in $v_1,\cdots,v_r,w$ ...
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3 answers
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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 $\overrightarrow{AB}=(5,2)=\...
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2 answers
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The proof of Triple Vector Products [duplicate]

THe question is to show the following: $$\vec{a}\times (\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$$ $$(\vec{a}\times \vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\...
Jason Ng's user avatar
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4 votes
4 answers
856 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 & ...
abouttostart's user avatar
9 votes
4 answers
31k views

Why does the magnitude of the cross-product of a and b give the area of a parallelogram spanned by a and b?

I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working ...
User001's user avatar
6 votes
1 answer
8k views

Cross product with curl

I'm trying to compute $u \times(\nabla \times v)$. My solution so far is below: \begin{align*} [u \times (\nabla \times v)]_i& = \epsilon_{ijk}u_j\epsilon_{klm}\partial_lv_m \\ & =\epsilon_{...
user374859's user avatar
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4 answers
2k views

Geometric understanding of the Cross Product

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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Find the equation of the plane knowing that it passes through 3 points

I have to find the equation of the plane that passes through $(0, 0, 0), (4, 0, -2), (0, 8, -6)$. I have done the following: The equation of the plane is of the form $$ax+by+cz+d=0$$ Since the ...
Mary Star's user avatar
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4 votes
2 answers
219 views

The identity $(u\times v)\cdot(x\times y)=\begin{vmatrix}u\cdot x&v\cdot x\\ u\cdot y&v\cdot y\\\end{vmatrix}$

I know by brutal calculation this identity holds always: $$(u × v) \cdot (x × y) = \begin{vmatrix} u \cdot x & v \cdot x \\ u \cdot y & v \cdot y \\ \end{vmatrix}$$ for arbitrary vectors $...
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3 votes
3 answers
942 views

Four dimensional cross product of THREE vectors

There are many MSE posts about how to define a cross product in $\mathbb{R^4}$. It is impossible to define a cross product of two vectors in $\mathbb{R^4}$, since there are infinitely many directions ...
Ma Joad's user avatar
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0 votes
1 answer
421 views

Vector cross product in cylindrical coordinates

I need to calculate the cross product of two vectors given in cylindrical coordinates but i can't find the formula for it anywhere online. Is it expected that i find the formula myself or is there ...
Manouil's user avatar
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17 votes
2 answers
13k views

Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
beep-boop's user avatar
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13 votes
8 answers
27k views

Visual Ways to Remember Cross products of Unit vectors? Cross-product in $\mathbb F^3$?

Objective to find visual and accessible ways to remember this formula fast $$(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)$$ I have used Sarrus' rule but it is slow, more here. Since it is slow, I have ...
hhh's user avatar
  • 5,477
11 votes
4 answers
16k views

Skew symmetric matrix of vector

During my course in linear algebra, the instructor stated that A cross B is the same as the "skew symmetric matrix" of A times B. So, first of all, can someone clarify or provide sources about skew ...
Abu Bakr's user avatar
  • 365
10 votes
3 answers
7k views

Geometric proof for triple vector product Jacobi identity

I believe the vector identity $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = 0$ is called the Jacobi identity and I know ...
Gerard's user avatar
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9 votes
6 answers
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Coordinate free proof for $a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$

The vector triple product is defined as $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})$. This is often re-written in the following way: \begin{align*}\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \...
Harambe's user avatar
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9 votes
2 answers
5k views

What is the interpretation of homogeneous line intersection?

I understand homogeneous coordinate systems. I read the intersection of lines in homogeneous coordinate can be computed by taking a cross products of lines $l_1(a_1,b_1,c_1)$ and $l_2(a_2,b_2,c_2)$. ...
Stack crashed's user avatar
7 votes
5 answers
4k views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
Norwolf's user avatar
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6 votes
3 answers
3k views

Differential equation involving cross product.

I have the differential equation $$f'=c \times f$$ for $f: \mathbb{R} \to \mathbb{R}^3$ and constants $c \in \mathbb{R}^3$. How can I solve something like this?
Dimtsol's user avatar
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5 votes
1 answer
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Orthogonal matrix over cross product

Let $a$ and $b$ be two unitary vectors in $\mathbb E^3$, and let $Q$ be an orthogonal matrix. Does the following hold? $$Qa \wedge Qb = \pm Q(a \wedge b)$$
user avatar
5 votes
3 answers
815 views

Cross-product identity

This page of vector identities lists the following (among many other identities): $$ (\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}))\,\mathbf{D}= (\mathbf{A}\cdot\mathbf{D} )\left(\mathbf{B}\times\...
Ben Grossmann's user avatar
4 votes
1 answer
11k 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}= \begin{pmatrix}a_{...
lambertmular's user avatar
3 votes
2 answers
2k views

on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$

I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct. First, when I actually multiplied out $\vec{B} (\vec{A} \cdot\vec{C})-\vec{C}(\vec{...
Jesse's user avatar
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3 votes
1 answer
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invariance of cross product under coordinates rotation

Question goes as If $\vec A$ and $\vec B$ are invariant under rotation, the prove that $ \vec A \times \vec B $ is also invariant. However solution of on the other page is not given. Says that if ...
S L's user avatar
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3 votes
2 answers
1k views

Define vector that forms equal angles to 3 other vectors

How should I define a vector, that has equal angles to vectors $\vec{i}, \vec{i} + \vec{j}$ and $\vec{i} + \vec{j} + \vec{k}$? After looking at the problem in a graphical way, I tried taking average ...
A. Smith's user avatar
3 votes
2 answers
1k views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
user10676's user avatar
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3 votes
5 answers
1k views

Help understanding cross-product

I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
Smashery's user avatar
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3 votes
1 answer
351 views

How to prove that a sum of quintuple cross products is equal to zero?

Show that : $$ p \times [(a \times q) \times (b \times r)] \\ + q \times [(a \times r) \times (b \times p)] \\ + r \times [(a \times p) \times (b \times q)] = 0 $$ where $\times$ is cross product and ...
Harendra Pratap SIngh's user avatar
3 votes
1 answer
7k views

How do you integrate Cross Products?

Hey I'm doing a course in mechanics and these keep cropping up! So for this question I'm working in 3d, and so far have $$m \mathbf{k} \cdot (\mathbf{q} \times \ddot{\mathbf{q}} )=0$$ so I need ...
Blue's user avatar
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2 votes
1 answer
276 views

Deriving formula for cross-product.

It is given on pg. #106, 107 in the book by: Thomas Banchoff, John Wermer; titled: Linear Algebra Through Geometry, second edn.. Consider a system of two equations in three unknowns: $$a_1x_1 + a_2x_2 ...
jiten's user avatar
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2 votes
1 answer
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Is there a 'geometric' version of this derivation of the vorticity equation?

Recall that for each vector $\omega\in\mathbb R^3$, there is an anti-symmetric matrix $ [\omega]_\times\in\mathbb R^{3\times 3}$ (and vice-versa) such that $$[\omega]_\times h= \omega\times h.$$ ...
Calvin Khor's user avatar