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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Orthogonal matrix over cross product

Is $Qa \wedge Qb = \pm Q(a \wedge b)$, where $a$ and $b$ are two unitary vectors in $E^3$ and $Q$ is an orthogonal matrix ??? Thanks
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If u(t) = < sin 6t, cos 6t, t >and v(t) = <t, cos 6t, sin 6t >, find d/dt u(t) × v(t) . [closed]

Hi I have been stuck on this problem for forever and can't figure out what I did wrong. Thanks! I got: <-6sin(6t)^2 -cos(6t)+6cos(6t)^2 +6tsin(6t), 2t-6cos(6t)sin(6t)+6sin(6t)^2, 6cos(6t)^2 + ...
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Volume of tetrahedron using cross and dot product

Consider the tetrahedron in the image. Prove that the value of the tetrahedron is given by $\frac16 |a \times b| \cdot c$ So far, what I did was I know volume of the tetrahedron is equal to the ...
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Simplification challenge (cross products)

In a calculation involving three 3D-vectors $A$, $B$, $C$, the following term appeared $$ 2\langle A\times B, C\rangle^2-\langle A\times B + B\times C + C\times A, (B\times C)\langle A, A\rangle + (C\...
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Given $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and $\vec{w}$ is $\perp$ to both find $\vec{u} \cdot \vec{v} \times \vec{w}$

I am given the following problem: Knowing that the angle between the unit vectors $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and that $\vec{w}$ is orthogonal to both of ...
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If $p \times q = 3p \times r$, then $q - 3r = \lambda p$

Three non-zero and non-parallel vectors $p,q$ and $r$ are such that $p \times q = 3p \times r$. Show that $q - 3r = \lambda p$, where $\lambda$ is a scalar. My attempt: We have $$p \times q = 3p \...
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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 ...
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How to tidy up this Cross Product derivation?

Last week I wrote this answer. However, I don't feel it is complete. I make an arbitrary choice of unit vector perpendicular to u,v. But I can't see how to integrate this fact into the derivation. ...
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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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Normal to surface at point

I have this function: $F(x,y,z)=x^2−y^2−z^2+4$ where $z\ge 0,0\le x \le 2,0 \le y \le 2$. How can I find the normal at some point $P=(p_x,p_y,p_z)$? I have tried to calculate the derivatives of ...
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420 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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4answers
347 views

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 ...
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1answer
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How do you find the max value of a length of a vector?

I have a vector $v = 7j$ and a vector $u$ with a length of 5 that starts at the origin and rotates in the $xy$-plane. How am I supposed to find the max value of the length of the vector $|u \times v|$...
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1answer
71 views

“Universal property” of cross product

Let $V$ be a three dimensional euclidian vector space which is oriented. Because of the orientation, we can define the cross product $\times: V^2 \rightarrow V$ uniquely by: $<v\times w,u> = \...
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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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2answers
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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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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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1answer
53 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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1answer
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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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2answers
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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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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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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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2answers
729 views

Interpretation of eigenvectors of cross product

If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ (...
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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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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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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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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} ...
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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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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
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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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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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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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Cross Product in 3D

Hi! I am currently working on some calc2 online homework problems concerning the cross product. I understand how the cross product works, but I am not sure how to apply it to this question. I know ...
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Vectors and Cross Product

I have these two questions regarding the Cross Product. 1.) You are looking down at a map. A vector $u$ with $|u| = 3$ points north and a vector $v$ with $|v| = 10$ points northeast. What is $|u \...
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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 is the matrix-defined Cross Product of two 3D vectors always orthogonal?

By matrix-defined, I mean $$\left<a,b,c\right>\times\left<d,e,f\right> = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$...
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1answer
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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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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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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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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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1answer
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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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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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1answer
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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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494 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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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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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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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 $D$. ...