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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Magnitude of torque due to weight in a simple pendulum

Suppose we have a simple pendulum as shown in figure . In this frame, suppose we fix $\theta$ as positive if rotation is at right of axis of symmetry (as depicted in figure) and negative if rotation ...
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How to intuitively understand cross-product

I have learned about the cross-product of vectors in both mathematical studies and physical ones but I am still curious as to how one can obtain an intuitive understanding of how two vectors create a ...
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Establishing a new coordinate sytem in a plane defined by 4 points

Suppose I am given 4 points in 3D in the global coordinate system [X,Y,Z]. Now, I would like to establish a new coordinate system [x',y',z'] such that o' is in the center of the plane defined by the ...
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Getting a wrong result while computing the cross product of unit vectors $\hat{i}$ and $\hat{j}$

Consider the coordinate system as shown below and unit vectors along x and y, $\hat{i}$ and $\hat{j}$. The cross product of these unit vectors as given here is equal to $\hat{k}$. However, when I ...
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Cross product implying vectors are equal [closed]

For 3-D vectors $a$, $b$ prove: $a \times b = a − b$ implies $a = b$ I've been working on this question for a while and have no idea how to solve it, any help would be greatly appreciated, thanks.
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How does an almost complex structure on a sphere determine a 1-fold cross product?

In the article "VECTOR CROSS PRODUCTS ON MANIFOLDS", by Alfred Gray, the author states, between theorem 2.8 and corollary 2.9, that The existence of vector cross product of Type I [1-fold ...
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Proving the cyclicity of scalar triple product

I wanted to proof the property that box product of three vectors remain equal if we change vectors in cyclic manner. i.e. $$(\vec{a}\times \vec{b}).\vec{c}=(\vec{b}\times \vec{c}).\vec{a}=(\vec{c}\...
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For which t values the two vectors are parallel to each other

$\overrightarrow{r_{1}}(t)=[t+6,-3,t+2]$ $\overrightarrow{r_{2}}(t)=[-10,t+7,-2t^{2}]$ For which t values the two vectors are parallel to each other? My try: I tried cross product, and got : $$\left[...
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Order of the cross-product preference $T_u \times T_v$ vs. $T_v \times T_u$

To explain this question better, I was working through my lecture's problem sets and this problem came up: Vector Calculus 6th Edition, Anthony Tromba, Jerrold E. Marsden Consider the closed surface $...
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Should a compactly supported field have a Helmotz decomposition that is compactly supported?

Let $\bf F$ be a smooth vector field, which is null outside a finite compact domain $V$. By Helmoltz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} ...
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Scalar triple product of three vectors at 60 degrees to each other

I encountered a question involving a scalar triple product of three vectors of unit length which were mutually at 60 degrees to each other. While the answer involved taking the root of the square of ...
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How does this code Find the intersection point between two lines?

I've been racking my brain for a while trying to step through this. This is Unity C# code used to find the position of intersection between two lines. The full function is here Because of my usecase I ...
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Problem understanding matrix product

In a textbook I have the following affirmation: $$\begin{bmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{bmatrix}^n=\frac{1}{0.3}\begin{bmatrix} 0.2 & 0.1 \\ 0.2 & 0.1 \end{bmatrix}+\frac{(0.7)^n}...
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Volume to surface integral of $R\times \nabla \times B$

I need to transform the following integral into a surface integral (if that's possible): $$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$ where $R = (...
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Relationship between cross-product and Moore-Penrose pseudoinverse [closed]

It is said in here https://blog.bham.ac.uk/intellimic/g-landini-software/colour-deconvolution-2/ that you can get the third vector of a 3x3 (stain) matrix either by taking the cross product of the ...
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Area of a parallelogram spanned by two 4D vectors without using trigonometry

In 3D, we can find the area of the parallelogram spanned by two vectors by using the cross product: $$Area = {\vert\vec a \times \vec b\vert}$$ In 2D, we can perform a similar operation using a 2D ...
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If the cross product result is known, then how to calculate the factors vector $A$ and vector $B\,$?

Assume that $$\nabla H_1\times\nabla H_2 \:=\: V\quad\text{and}\quad V \:=\: \big(σy, x(r − z), xy\big)\,.$$ My question: If $V$ is given, is there any way to find out what $\,\nabla H_1\,$ and $\,\...
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Finding the conditions for $\mathbf{A}×(\mathbf{B}×\mathbf{C}) = (\mathbf{A}×\mathbf{B})×\mathbf{C}$.

I wanted to find the conditions in which $\mathbf{A}×(\mathbf{B}×\mathbf{C})$ is equal to $(\mathbf{A}×\mathbf{B})×\mathbf{C}$. On solving $\mathbf{A}×(\mathbf{B}×\mathbf{C})-(\mathbf{A}×\mathbf{B})×\...
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inverse action of cross product

Suppose we have two vectorsfield A(x,y,z) and B(x,y,z).If we know B = (0,2,1) can we compute A if: ? EDIT: After some searching online I found that there are infinitely many vectors fields A(x,y,z) ...
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Normal vector to a plane containing the Origin

The points A,B,C have position vectors; 2i+2j, -j+k and 2i+j-7k respectively, relative to the origin. Find the normal vector to the plane OAB. I know that $$\vec {OA}$$ and $$\vec {OB}$$ are in the ...
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An example of integral with cross product

Suppose I have a uniform vector field ${\mathbf{f}}({\mathbf{x}}):\mathbb{R}^3 \to \mathbb{R}^3$. Let us fix a cylindrical coordinate system such that ${\mathbf f}({\mathbf x})=f\,\hat z$, where $f$ ...
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Where did I go wrong computing the cross product

If $u=(-2,0,\sqrt2)$ and $v=(0,-4,-\sqrt2)$, then $u\times v=(2\sqrt2,-\sqrt2,8)$. I got $(4 \sqrt2, 2\sqrt2,0)$ for the cross product but the answer is saying I'm wrong and I don't know why
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Cross product: does Jacobi identity imply Lagrange identity?

A cross product can be defined as a bilinear operation on a real vector space with inner product that has the property of orthogonality: $ \mathbf u \cdot(\mathbf u \times \mathbf v)=\mathbf v \cdot(\...
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What type of product? $\mathbf{a} \circledast \mathbf{b}:= \mathbf{a}\mathbf{b}^T - \mathbf{b}\mathbf{a}^T$

What do you know about the names and properties of the products of the following definitions? \begin{align} &\mathbf{a,b}\in\mathbb{R}^N, \mathbf{C}\in\mathbb{R}^{N\times N}\\ &\mathbf{C} = \...
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Vector Product of complex vectors

I need to get a vector orthogonal to $\mathbf{A}=(1,i,1,-i)$ and $\mathbf{B}=(i,1,i,-1)$ where $i^2=\sqrt{-1}$. Note also that $\mathbf{A}.\mathbf{B}=0$. I was thinking to take the cross product, but ...
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If A and B are two given vectors and what is the procedure to find A.▽B and A x ▽B at a given point?

This is what I did: Calculated ▽B and substituted the value of the point to find a vector Calculated A at the given point by substituted the coordinates in the vector Dot product of the two to find ...
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Finding the value of the cross product of 2 vectors without knowing the value of the vectors?

Let this be the question: Suppose $\vec{v}$ and $\vec{w}$ are two vectors parallel to the plane $$x + 2y + 3z = 7.$$ Suppose furthermore that $\vec{v}$ is perpendicular to $\vec{w}$, $$‖v‖= 3, \ ‖w‖= ...
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Proving result for vectors $\bf u_1,u_2,v_1$ and $\bf v_2$ [closed]

Question For vectors $\bf u_1,u_2,v_1$ and $\bf v_2$, Show that $$\begin{vmatrix}\bf{u_1\cdot v_1} & \bf{u_1\cdot v_2} \\\bf{u_2\cdot v_1} & \bf{u_2\cdot v_2} \\\end{vmatrix}=(\textbf{u}_1\...
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Autocad WCS to OCS conversion (rotational matrices)

Trying to convert an objects coordinates from WCS to OCS. I am only interested in the mathematics of this problem and not using an existing library to do this for me. I have an object that exists with ...
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Intuition: Equation of a Plane by three points (position)

For three arbitrary points, $A$, $B$, $C$, with position vectors $\bar a, \bar b, \bar c$, the equation of the plane can be expressed as: $$(\bar a \times \bar b) \hat x + (\bar b \times \bar c) \hat ...
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Disproving the statement A×B⊆C×D if and only if A⊆C and B⊆D

I know that a counterexample can disprove this statement I tried a lot to disprove it without using a counterexample If A⊆C and B⊆D let (x,y)∈ (A×B) ⇒x∈A and y∈B ⇒x∈C and y∈D(from the assumption) ⇒A×...
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Vector product: What came first? The method, or the need for the result?

I'm reviewing vector products, and I was always just taught here's how we calculate the vector product, and here's a list of properties that it satisfies (orthogonal to the plane spanned by the ...
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How to solve $\ddot{\vec{u}} = \vec{u} \times \hat{k}$

How do I solve?: $$\ddot{\vec{u}} = \vec{u} \times \hat{k}$$ I have tried solving a simpler version of this, $\dot{\vec{u}} = \vec{u} \times \hat{k}$. This one was easy: the head of the vector rotates ...
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Good notation for perpendicular vectors?

I have a unit vector $\mathbf{\hat{a}}\in\mathbb{R^3}$. I would like to calculate two new unit vectors $\mathbf{\hat{b}}$, $\mathbf{\hat{c}}$ which are perpendicular to $\mathbf{\hat{a}}$ and to each ...
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Differential equation involving cross product with periodic solution

Consider a system of differential equations $$ \begin{cases} \dot{p}(t)= \frac{1}{R} p(t) \times q(t)\\ \dot{q}(t)= - p(t) \times q(t) \end{cases} \qquad (1) $$ for $p(t),q(t)\in \mathbb{R}^3$. Of ...
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Finding a specific dot product from a cross product

I am doing the no bullsh*t guide to linear algebra. On page 161 problem 2.9 is: Find a vector that is orthogonal to both $u_1 = (1,0,1)$ and $u_2 = (1,3,0)$ and whose dot product with the vector $v = ...
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Cross Product: is there an inverse?

I learning vector calculus and electromagnetism, from which I noticed cross product is used extensively. I.e. $\phi = \Delta \times \mathbf E$. I'm curious, is there an inverse to the cross product? ...
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The importance of compactness in the Tube Lemma [closed]

I know that to use the Tube Lemma at least one of the spaces must be compact, but if none of the spaces are compact, then how can I find a counter example? The Tube Lemma Let X be a topological space ...
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How do you prove the equivalence for algebraic and geometric definition for both dot and cross product?

How do the algebraic and geometric formulas of dot and cross product relate to each other? I have seen the proof for 2 dimensions but how do we generalize it to $n$ dimensions? Is there a way to prove ...
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Rodrigues equation VS Hamilton's quaternions. Historical confusion.

I don't have a special math education and when I study quaternions I spent a long time. Along the way, without realizing, I independently derived the Rodrigues equation, because now we know about the ...
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4 votes
1 answer
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How close can we get to a cross product in dimensions other than $0, 1, 3, 7$?

A cross product is a bilinear operation which, given two input vectors $x, y$, produces a vector $x \times y$ orthogonal to both, whose length equals the area of the parallelogram spanned by $x$ and $...
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How can a Cross-Product give a vector solutions is missing [closed]

I am trying to understand the function of a cross-product If you take the cross-product of V x U gives a another vector that is pendicular say call it c-vector? V * c-vector = 0 U * c-vector = 0 But ...
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$\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot ( \vec{c} \times \vec{a})$?

I've seen my teacher of general physics write $\vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot ( \vec{c} \times \vec{a})$ but I've search for a proof in google and there is nothing. I also ...
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Least Squares Solution with Cross Products

Is there a way to find a least squares solution for a vector using a system of cross product equations? For example $\vec{A}, \vec{B}, \vec{C}, \vec{D}$ are all known quantities in 3D space: $$\vec{A} ...
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Can it be true that $\|v^t x\|>1$?

Let $\|v\|=1$ and $v\in R^n$. Let $x = (1,-1,1,1,-1,....,1)\in R^n$ ($x$ is just a vector of $1$'s and $-1$'s. Can it be true that $\|v^t x\|>1$? I'm looking for examples or a proof that it can't.
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Formality of Units in Cross Product

I recently lost some marks for answering a question on an assignment about the cross product of two lengths. I listed my answer in units of cm, instead of in units of cm^2 and that got me docked. This ...
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How to determine if a segment passes through the positive X axis?

A segment is defined by two endpoints $(x_1, y_1)$ and $(x_2, y_2)$. The positive X axis for simplicity can be defined as the segment going from $(0, 0)$ to $(10^9, 0)$ (I'm working on a program where ...
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3 votes
3 answers
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Using cross product find direction vector of line joining point of intersection of line and plane and foot of perpendicular from line to plane.

A line with equation $r=a+\lambda\vec{d}$ meets plane $\pi$ with equation $r.\hat{n}=k$ at point P. Point Q lies in $\pi$ and is the foot of the perpendicular from A to $\pi$. Find the direction ...
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ODE with cross products

I've been working on a project about the dynamics of spins and I've encountered the following system of ODEs which I'm unable to solve: $$\frac{dS_1}{dt} = S_1 \times S_2 + \gamma S_1 \times (S_1 \...
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2 votes
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Generalizing the result $|a\times b|=|a||b||\sin \theta|$ to arbitrary dimensions.

The cross product can be generalized to arbitrary dimensions as done below or here. I'm trying to state and prove the general analogue (for arbitrary dimensions) of the equation $$|a\times b| = |a||b||...
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