For questions on cross products.

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2
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1answer
513 views

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 ...
-1
votes
2answers
24 views

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 ...
0
votes
0answers
15 views

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} / ...
24
votes
1answer
1k views

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 ...
5
votes
2answers
284 views

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 ...
0
votes
0answers
17 views

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 : ...
4
votes
2answers
718 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 $\boldsymbol{x}_1=t\boldsymbol{v}$ ...
1
vote
0answers
14 views

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 ...
0
votes
0answers
22 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
votes
2answers
34 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 ...
4
votes
8answers
4k 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} ...
0
votes
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
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 ...
2
votes
2answers
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
0
votes
1answer
52 views

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 ...
1
vote
2answers
437 views

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 ...
0
votes
0answers
31 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 $\mathbb{R}^3$. ...
0
votes
1answer
16 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
10
votes
4answers
1k views

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} ...
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
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 $l_2(a_2,b_2,c_2)$. ...
1
vote
3answers
42 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
vote
2answers
41 views
1
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2answers
43 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
vote
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
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
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
1answer
472 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 ...
0
votes
2answers
46 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
26 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
votes
2answers
54 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
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?
0
votes
1answer
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 ...
1
vote
1answer
28 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
votes
0answers
24 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
70 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
vote
1answer
26 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
39 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 ...
29
votes
4answers
6k views

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 ...
0
votes
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
votes
4answers
101 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 ...
1
vote
2answers
171 views

Cross Product for Biot-Savart Derivation of Current Loop

Biot-Savart's law can be used to determine the magnetic field produced by a figure at a point. Introductory physics texts integrate $dB$ to obtain $B$ where $$dB = \frac{I\mu_{0}}{4\pi r^2} dl ...
1
vote
2answers
127 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
1
vote
2answers
85 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.