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 difference between cross product and exterior product?

I have learn that the exterior product is an oriented plane called bivector given as $A \times B = |A||B| \sin x (i \times j)$ For $x \in(-\pi,\pi)$. I will like someone to derive the cross product ...
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49 views

How to solve proportions involving vector cross products?

I have the following proportion $\vec{JV} \times \vec{F_v} = \vec{JM} \times \vec{F_m}$ and all members are known except the magnitude of the vector $\vec F_m$, like described by another question here ...
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65 views

Is the angle between a and b is equal to the angle between b and a?

This was a question in an exam: Calcualte tan of the angle between a and b if:a = (4,3) and b = (5, -12) There are two answers to this question: Some students devided the dot product of a and b by ...
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32 views

vector and curl identity

This popped up in my notes and the author made no remarks about the properties used $\bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0}$ Then, $\...
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Finding a plane perpendicular to two lines and a point

A plane is perpendicular to both [x,y,z] = [1, -10, 8] + s[1, 2, -1] and [x,y,z] = [2, 5, -5] + t[2, 1, -3], and contains the point P(-1, 4, -2). Determine if the point A(7, 10, 16) is also on this ...
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How come the angle for cross and dot equations are not equivalent in these equations?

Finding the angle between two lines Given lines: $$l_1 = [3, 1, -1] + t[2, -2, 3]$$ $$l_2 = [5, -1, 2] + t[1, -3, 5]$$ I tried cross product equation of finding the angle: $\cos^{-1}(\sqrt{426} / (\...
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Edge Chromatic Number of Product Graphs

Assume that two graphs like $G$ and $H$ are given. $G \times H$ is a graph such that every vertex of it comes from $V(G) \times V(H)$ and every vertex like $(u,v)$ is adjacent to $(u',v')$ iff : $1$...
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Geometric interpretation for eigenvalues and eigenvectors of the cross product's representation as a linear map

Fix ${\bf x} = (x_1,x_2,x_3) \in \Bbb R^3\setminus\{{\bf 0}\}$. We can look at the cross product as a linear map ${\bf x}\times: \Bbb R^3 \to \Bbb R^3$ which is represented in the standard basis by $$\...
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rearranging integral of cross product

I am given the integral \begin{gather} \int_V \hat{e}_z \times \vec{u} dV \end{gather} where $\hat{e}_z$ is the unit vector in the $z$ direction and $\vec{u}$ is a vector field. Can I pull $\hat{e}...
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23 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 $(\bf a \times \bf b) \dot \bf c$?
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41 views

What determines the direction of cross product resultant vector?

Why do we use the right hand rule to determine the direction of the vector resulting from using the cross product? A resultant vector that was directed in the opposite direction would also be ...
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30 views

Cross product angle formula

Say there are two vectors $A$ and $B$ in $3D$. to get the angle between the cross product of those two vectors, you use $$||A\times B|| = ||A||\;||B||\sin(\theta). $$ right? Is this equation ...
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24 views

Proof of $(A+B) \times (A-B) = -2(A X B)$ [closed]

Proof of $(A+B) \times (A-B) = -2(A \times B)$, where 'A' and 'B' are vectors
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1answer
59 views

Show that ∇· (∇ x F) = 0 for any vector field [duplicate]

To solve this question, how do I define any vector field $F$, in order to solve it? I called $F = (ax,by,cz)$, in which case already $\nabla\times F = 0$. How would i go about proving this? Many ...
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36 views

Which sides of a triangle are visible to an observator?

Working o 2-d plane. Supposing that there is a observer standing on the origin (0, 0) looking to the first quadrant. If there is a triangle drawn on the first quadrant, what sides are visible to the ...
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29 views

Cross products and orthogonal complements

I am having trouble with this question about cross products and orthogonality: Let a ∈ R3 \ {0} Show that if y ⊥ a then $\exists$ x {x ∈ R3 : a × x = y} Could anyone explain this to me? Thanks
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18 views

How to show this vector cross product/gradient result

One of my books has that if $$\bar A= \phi \nabla \psi$$ then $$\nabla \times \bar A = \nabla \phi \times \nabla \psi$$ But I don't see why it is true. What is the proof of this? Thanks
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Why and how two skew vectors' cross product gives normal vector of plane containing one of those vectors

I got a question which says : Given $$\vec{v} = <1,0,-1> $$ and line $$L_1 : (1-2t)\vec{i}+(4+3t)\vec{j}+(9-4t)\vec{k}$$ Find an equation of plane $P$ which is parallel to the vector $\vec{v}...
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42 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 $l_2(a_2,b_2,c_2)$. ...
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46 views

Solving vectors such that the dot product = 0

I'm doing some machine learning problems (namely logistic regression), and something I'm trying to do is calculate the decision boundary given a weight vector $\mathbf{w}$. The decision boundary lies ...
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41 views

How to solve triple cross product?

I have no idea , how to start. And: 5π/6
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Understanding the definition of the cross product

I know how to use the cross product, I know what it means and how it relates to the dot product. $$|a \times b| = ||a||b| \cdot \sin(\theta) \vec{n}|\\ a \cdot b = |a||b| \cdot \cos(\theta)$$ I ...
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Biot-Savart Law to construct vector potential for divergence free vector field on $\mathbb{R}^3$

I would like to confirm a method I am trying to use which uses the Biot-Savart Law to construct a vector potential $\underline{w}$ for a divergence free vector field $\underline{v}$ on $\mathbb{R}^3$. ...
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54 views

Matrix product demonstration

Sorry for boring you my friends. I have haunted by a problem of relation between matrix product and cross product. I would like to demonstrate the following equation: $$ (\Omega\cdot r)^T(\Omega\cdot ...
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1answer
15 views

Show that if $Q'$ is any point on the line of action of $F$, then $PQ × F$ = $PQ'× F$

If a force $F$ is applied to an object at a point $Q$, then the line through $Q$ parallel to $F$ is called the line of action of the force. We defined the vector moment of $F$ about a point $P$ to be $...
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56 views

Find if a vector is between 2 vectors [duplicate]

I have a label which is linked to an anchor. The problem is to find on which one of the four side of the label (which is a Rectangle) should be linked to the anchor. ...
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50 views

Arbitrary Dot and Cross Products

I am having a bit of trouble with answering these few dot and cross product questions. Suppose that $u · (v × w) =3$. Find, $w · (u × v)$ $v · (u × w)$ $(u × w) · v$ Could some explain their ...
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23 views

Showing the distance between a point $P$ the line determined by a segment $AB$ is $d=\frac{||AP\times AB||}{||AB||}$

Show that in $3$-space the distance $d$ from a point $P$ to the line $L$ through points A and B can be expressed as $$d=\frac{||AP\times AB||}{||AB||} .$$ My diagram of the situation: My next ...
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31 views

Logic behind cross and dot products

Let $A, B, C, $and $D$ be four distinct points in $3-space$. If $AB×CD$ does not equal $0$ and $AC⋅(AB×CD)=0$, explain why the line through $A$ and $B$ must intersect the line through $C$ and $D$. ...
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24 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}...
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61 views

Determining an unknown vector from its cross and dot product with known vector

Let $\vec{k}$, $\vec{v}$, and $\vec{u}$ be vectors, where $\vec{u}$ is unknown and $\vec{k}$ and $\vec{v}$ are known vectors. Given: $\vec{u}\cdot\vec{k}=c$ $\vec{u} \times \vec{k}= \vec{v}$ ...
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25 views

Find the magnitude/length of the cross product of two vectors [duplicate]

I'm going through past exam questions, and this is one I haven't come across. How can I approach it?
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21 views

Order of operations - cross product and simple multiplication [closed]

I'm just wondering which takes precedence or if it really matters. It would matter wouldn't it? For example, this is written in my textbook: Equation for magnetic field of a point charge so the [qv ...
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1answer
30 views

Scaling by Jacobian for cross product?

I am trying to show that if $X:U\to\mathbb{R}^3$ is a parametrization of a coordinate patch on a refular surface $S$ and $F:U'\subset\mathbb{R}^2\to U$ such that $Y=X\circ F$ is a regular ...
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Integrating a cross product by parts

I'm looking to integrate the following using something like integration by parts and am not exactly sure how to go about it/where to begin. $\int (w \times u) \cdot v $ Where $w= \nabla \times u$ ...
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Intuition behind cross-product and area of parallelogram

The cross product in 2D is defined like that: $|(x_1, y_1) \times (x_2, y_2)| = x_1 y_2 - x_2 y_1.$ I perfectly understand the first part of the definition: $x_1 y_2$, which is simply the area of a ...
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28 views

Area of a parallelogram using cross product, how can length be equal to area?

We get a vector by a cross product and its length (magnitude) is the area of the parallelogram. How is this possible as the unit of length is meters and unit of area is meters squared?
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41 views

Can one define a cross product for functions?

The dot product $c = \sum_i a_i b_i$ can be easily be generalized for continuous functions like $$ c = \int_{-\infty}^{\infty} a(x) b(x) d x $$ But can one also generalize the cross product $c_{ij} = (...
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Question about cross product of images of linear transformation

I'm reading "Differential Geometry: Curves and Surfaces" of Manfredo Carmo, and this part in the book confuses me(page 166): Suppose that $N: S \rightarrow S^2$ is the Gauss map of regular surface ...
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35 views

A proof to a vector identity

I do not know how to prove this, can anybody help me out with that? Consider five vectors: $\vec{a},\vec{b},\vec{c}, \vec{p}, \vec{q} \in \mathbb{R}^3$ then: $$(\vec{p}\cdot\vec{q})(\vec{a}\cdot(\...
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How does computing the determinant of a matrix with unit vectors give you the Cross Product?

Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...
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86 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
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89 views

How can we determine if two vectors are parallel?

What are the minimal number of products like dot cross that can give us information if two vectors are parallel ? What can we say if V*W = 1 assuming V and W are not unit vectors.
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How to calculate one of the vectors that generate a given cross-product?

Given the vector: $$\vec b=(-0.361728, 0.116631, 0.924960)$$ and it's cross-product: $$\vec a \times \vec b=(-0.877913, 0.291252, -0.380054)$$ How do I calculate $\vec a$ ? It's been a while since I'...
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Lack of associativity of cross product vs associativity of the exterior product

Can someone remind me in a nutshell why the associativity of the exterior product fails to transfer to the cross product? (It's been over a decade since I had to deal with the former back in school.) ...
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70 views

Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
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27 views

Homology of $T^3$ generated by three copies of $T^2$

Why is the second homology of $T^3=S^1\times S^1\times S^1$ generated by $S^1\times S^1\times \{\mathrm{pt}.\}$, $S^1\times \{\mathrm{pt}.\}\times S^1$ and $ \{\mathrm{pt}.\}\times S^1\times S^1$? $...
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23 views

Decompose cyclic sum of crossproducts into two cyclic sums?

Suppose you have $6$ points $a_i\in\mathbb{R}^3$ $i\in\{1,..,6\}$ such that all triangles with vertices $0, a_i, a_{i+1}$ for $i\in\{1,..,5\}$ do not degenerate (I dont know if this assumption is ...
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Finding the exact value of the sine of the angle between a line and a plane

I have done part (a). For part (b), I know the principle of how to do it, I tried to use the cross product to find the exact value of the sine of the angle. So I found PQ which is $$\begin{pmatrix}7-...