For questions on cross products.

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-4
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1answer
23 views

Vector cross product proof vs scalar [on hold]

In the set of $\mathbb{R}^3$ vectors $u,w,v$ does $u \times (v \times w) = (u\cdot w)v-(u\cdot v)w$ ? Note the difference between $\times$ and dot. Hint cross product.
0
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0answers
20 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 $(\bf a \times \bf b) \dot \bf c$?
0
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2answers
30 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 ...
0
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1answer
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 ...
-3
votes
1answer
22 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
1
vote
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 ...
0
votes
2answers
34 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 ...
2
votes
2answers
27 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
-1
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0answers
11 views

Find a 2-dimensional vector that is guaranteed not parallel to a finite set of 2-dimensional vectors?

Given a finite set of 2-dimensional vectors $V$, find any vector $\vec{x}$ such that if $\vec{v} \in V$ then $\mid \vec{x} \times \vec{v} \mid \ne 0$.
0
votes
1answer
13 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
0
votes
1answer
18 views

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 ...
2
votes
2answers
39 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 $l_2(a_2,b_2,c_2)$. ...
1
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3answers
41 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 ...
1
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2answers
39 views

How to solve triple cross product?

I have no idea , how to start. And: 5π/6
0
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2answers
40 views

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 ...
1
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0answers
30 views

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 $\mathbb{R}^3$. ...
1
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1answer
53 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 ...
0
votes
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 ...
0
votes
3answers
55 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. ...
0
votes
2answers
45 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 ...
0
votes
1answer
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 ...
1
vote
1answer
24 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 $D$. ...
0
votes
1answer
22 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} = ...
0
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2answers
40 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: $\vec{u}\cdot\vec{k}=c$ $\vec{u} \times \vec{k}= \vec{v}$ ...
1
vote
1answer
23 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?
0
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1answer
20 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 ...
1
vote
1answer
25 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 ...
0
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0answers
23 views

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 $w= \nabla \times u$ ...
3
votes
4answers
63 views

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 ...
1
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1answer
24 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?
0
votes
1answer
37 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} = ...
-1
votes
1answer
28 views

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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2answers
55 views
0
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1answer
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: ...
3
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4answers
92 views

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 ...
0
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2answers
73 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 ...
1
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2answers
81 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.
0
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2answers
50 views

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 ...
-1
votes
1answer
53 views

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.) ...
2
votes
1answer
63 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 $2^n - ion$ ...
0
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0answers
25 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$? ...
0
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0answers
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 ...
0
votes
4answers
53 views

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 ...
1
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2answers
68 views

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 ...
6
votes
2answers
403 views

Proof for vectors involving cross and dot product

Prove that for any two vectors $\mathbf a$ and $\mathbf b$, $\lvert \mathbf a \times \mathbf b \rvert^2 + (\mathbf a \cdot \mathbf b)^2 = \lvert \mathbf a \rvert^2 \, \lvert \mathbf b \rvert^2$. ...
0
votes
1answer
37 views

Is this a correct identity for the Kronecker delta and the Alternating Tensor?

If $\varepsilon_{ijk}$ is the alternating tensor and $\delta_{in}$ is the Kronecker delta, am I correct in thinking that $$ \delta_{in}\varepsilon_{ijk} = \varepsilon_{ink} $$ If not, what is the ...
9
votes
2answers
114 views

On defining cross (vector) product.

This has been bugging me for years so I finally decided to "derive" (for lack of a better term) the definition of the cross product in $\mathbb R{^3}$. Here was my method for finding a vector: ...
6
votes
1answer
81 views

Slick proof of cross product identities

The cross product between vectors in $\mathbb{R}^3$ obeys two pleasant identities (sometimes named after Lagrange), namely $a\times(b\times c)=b(a\cdot c)-c(a\cdot b)$ $(a\times b)\cdot(c\times ...
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0answers
27 views

Construct 3D plane from 2 points and minimize angle of two vectors with its normal

I have as input two points $P, Q \in E^3$ and two vectors $\vec{v}_1, \vec{v}_2 \in R^3$. I need to construct a plane $(\vec{n}, d)$ such that the two points are in the plane and the angles between ...
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2answers
36 views

How do I evaluate the following vector product

I need to evaluate the following $$((\vec{a}\times\vec{b})\times\vec{a})_i((\vec{a}\times\vec{b})\times\vec{a})_j.$$ I am assuming Levi civita notation would be useful, but couldn't utilise it. Does ...