For questions on cross products.
3
votes
1answer
31 views
Do the BAC-CAB identity for triple vector product have some intepretation?
As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
0
votes
1answer
32 views
Cross product as factor in dot product [duplicate]
Given there are two vectors $w,v$ with $||w||=4$ , $||v||=1$ and $\phi=\frac{2\pi}{3}$
How do you transform the following expression into a form in which it can be computed with the given ...
0
votes
2answers
59 views
Computing cross product using norm and angle
Sorry for the weird title, if someone finds a better title for my problem be my guest to edit it ;)
For $\mathbf{v,w} $ in R³ with $\mathbf{||v||=1 ;||w||=4; \theta
=\frac{2\pi}{3}}$
Solve ...
1
vote
1answer
30 views
Find vectors vertical to given vectors with certain length
Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are
vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1
Every vector $\mathbf{x'}$ that is to be found must meet these ...
1
vote
3answers
39 views
Reordering vector product
If I have vectors $a, b, c \in \mathbb{R}^3$, and if we have e.g. $a = b\times c$, is there any way to express $b$ in terms of the other two?
2
votes
2answers
54 views
Help over the proof of triple vector product identity
For all vectors $\bf{x}$, $\bf{y}$ and $\bf{z}$,
$$\bf{x}\times(\bf{y}\times\bf{z})=(\bf{x}\cdot\bf{z})\bf{y}-(\bf{x}\cdot\bf{y})\bf{z}$$
The proof goes as follows:
We may suppose that $\bf{y}$ ...
4
votes
1answer
99 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 $\boldsymbol{x}_1=t\boldsymbol{v}$ ...
4
votes
1answer
41 views
How to solve cross-products including matrices?
I'm a programmer and I'm doing a whitebalance-transformation in RGB colorspace. This should work with this transformation matrix that I've found in literature:
$$
\begin{pmatrix}
R \\
G \\
B
...
2
votes
3answers
33 views
Cross product- square
I recently saw the following expression somewhere-
$$\frac{1}{2} \left\| \frac{\vec{u}}{9} \times \frac{\vec{u} + \vec{v}}{9} \right\| + \frac{1}{2} \left\| \frac{\vec{u} + \vec{v}}{9}\times ...
2
votes
1answer
31 views
Cross product, ortonormal basis
Could you explain to me why for $\{i, \ j, \ k\}$ an orthonormal basis of $\mathbb{R}^3$ we have $i \times j =k, \ \ j \times k = i, \ \ k \times i =j$?
Thank you.
1
vote
1answer
40 views
Comparing a geometric definition of cross product to the “usual” one
Could you help me with my little problem?
Given this definition of cross product:
1) $a \times b$ is perpendicular to $a$ and $b$, whenever $ a,b$ are linearly independent
2) basis $a, \ b, \ a ...
1
vote
1answer
51 views
Solution set to cross product
If $\vec a,\vec b \in \mathbb{R}^3$ with $|\vec a|\ne0$ show that the equation $\vec a \times \vec u =\vec b$ has a solution if and only if $a \cdot b = 0$ and find all the solutions in this case.
...
1
vote
2answers
35 views
Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is maximal, and then a unit vector $w$ s.t $|(u\times v)\cdot w|$ is minimal
This is a similar question to the one I have posted before. The problem
is as in the title:
Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is
maximal, and then a unit vector $w$ s.t ...
-4
votes
2answers
120 views
Why cross product's formulas defined in this way?
Why cross product's formulas defined in this way?
When mathematicians need to define cross product?
2
votes
1answer
35 views
Cross Product for functions
So functions are just uncountabley-infinite dimensional vectors, and as such there's a nice generalization of the inner product between two functions (the integral of their product). Is their a ...
1
vote
3answers
247 views
Area of a parallelogram (linear algebra)
Find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.
1
vote
1answer
46 views
Test of handedness
I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors.
if (v1 x v2) . v3 > 0 then it's right-handed, while if ...
1
vote
2answers
55 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 ...
1
vote
1answer
80 views
How to generate an ordered list of vertices of a cube from a face and a normal vector
Consider a cube with faces we'll call "left", "right", "front", "back", "top" and "bottom".
The cube can be described by $0 \le x,y,z \le 1$.
To name the faces, we'll say $x$ extends to the right, ...
0
votes
1answer
116 views
The Darboux vector is defined by $D = \tau T + \kappa B$. Show that $T' = D \times T$
The Darboux Vector is defined as $D = \tau T + \kappa B$. Show that for a unit speed curve
$$T' = D \times T \hspace{1cm} ... $$
Here, the $...$ represents the fact that there are a few ...
7
votes
1answer
117 views
Maps of $\mathbb{R}^3$ preserving the cross product
Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$:
$$\phi(a \times b)=\phi(a) \times \phi(b)$$
Is $\phi$ necessarily a rotation around the origin or the map ...
0
votes
1answer
35 views
Special Case of Lie-algebra
Suppose $\Bbb{R}^3$ with $[u,v]=u\times v$, thus the cross product of $u$ and $v$ and suppose also $\mathfrak{so}(n)$, the space of skew symmetric $n\times n$-matrices with $[a,b]=ab-ba$. Then i have ...
-1
votes
2answers
83 views
Find the sine sign given a pair of 3D vectors
I want to find the exact sine between two vectors in 3-dimensional space.
Data:
$x$: vector
$y$: vector
$z = \Vert x \times y \Vert$
I have tried this:
$$\sin \alpha = \frac{\Vert z\Vert}{( ...
4
votes
3answers
110 views
Rotational invariance of cross product
Hi guys I'm looking for a proof that $ ( Ra \times Rb ) = R ( a \times b ) $ where $\times$ is the three-dimensional cross product, and $R$ is a rotational matrix ( $\det R = 1$ and $R^T R = I$ )
...
2
votes
0answers
113 views
Cross product in higher than 3 dimensions
As I understand it, to get an $n$-dimensional cross product, you need $n-1$ vectors of dimension $n$. However my lecture notes are quite miss leading in the fact that they suggest this isn't always ...
1
vote
2answers
72 views
What does the symbol $\Delta$ stands for?
While studying Landau-Lifshitz equation following term appears,
$-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$
In above equation m is a vector quantity. It will be great if someone can ...
1
vote
1answer
56 views
Vector product proof
Prove that if
$$a=b \times c$$
$$b=c \times a$$
$$c=a \times b$$
then $a \perp b$, $a \perp c$, $b \perp c$, and $|a|=|b|=|c|=1$
0
votes
1answer
155 views
Cross product proof
Three vectors $a$, $b$, and $c$ are given. Prove that if $a \perp b$, $a \perp c$, $b \perp c$, and $|a|=|b|=|c|=1$, then
$$a=b \times c$$
$$b=c \times a$$
$$c=a \times b$$
1
vote
2answers
2k views
What is the general formula for calculating dot and cross products in spherical coordinates?
I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
0
votes
1answer
69 views
Cross product as result of projections
The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of ...
1
vote
1answer
117 views
proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$
I encounter this triple product property in wikipedia
But I can't find proof for
$$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$
The RHS cross ...
1
vote
1answer
106 views
Cross product and inverse of a matrix
I would like to show that $\left(\begin{array}{ccc}
1 & s & s^2 \\
1 & t & t^2 \\
1 & u & u^2
\end{array}\right)$ has an inverse provided $s$, $t$ and $u$ are distinct.
I ...
0
votes
1answer
42 views
Derivative of a vector function
Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus.
Find the derivative of the vector function:
...
0
votes
1answer
41 views
problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
can anyone expand on how the final answer is derived?
I try to expand but ended up scratching my head.
the best I can ...
2
votes
2answers
127 views
How to prove the equality of two vectors?
OK, i am trying to prove that if $\vec a\times \vec b = \vec a \times \vec c$
and also $\vec a\cdot \vec b = \vec a \cdot \vec c$ then $\vec b = \vec c$.
so far i got to $\vec n \tan \alpha = \vec m ...
1
vote
2answers
131 views
How do you compute the normal vector to a hyperplane in $\mathbb{R}^n$ given $n$ representative points?
Given $n$ points (no two identical, no three colinear, no four coplanar, etc.), I'd like to find a formula for the normal vector to the unique hyperplane that intersects each of these points.
In ...
6
votes
2answers
109 views
Explanation of a cross product result
In my book the result $$(u\times v)\cdot(x\times y)=\begin{vmatrix} u\cdot x & v\cdot x \\u \cdot y & v \cdot y\end{vmatrix},$$ where u, v, x and y are arbitrary vectors, is stated (here ...
-1
votes
3answers
60 views
Why is this not possible?
Why is the following not possible?
$$\frac{2x-1}{2x}\neq4x-2$$
And the following method not correct?
$$\bigg(\frac{2x-1}{2x} + \frac{1}{1}\bigg)-1\equiv\frac{2x-1}{2x}$$
Cross multiplying:
...
2
votes
1answer
234 views
Determining partial derivatives and cross products for bicubic interpolation using function values only?
I'm trying to implement a bicubic interpolation algorithm. In order to calculate the interpolated values, I need to calculate sixteen coefficients used in the calculation process - and that's where ...
0
votes
1answer
65 views
Another Cross Product
So I understand most of the properties of cross products. However I ran into a small complication.
I get that $i\times j = k$, $j\times k = i$. I also understand that $k \times j = -i$ and that ...
9
votes
4answers
655 views
Is the vector cross product only defined for 3D?
Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as
$$
\vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n
$$
It then mentions that $\vec n$ is the vector normal ...
2
votes
2answers
60 views
Cocountable fibers
Let $C$ be an uncountable set.
Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
2
votes
1answer
317 views
invariance of cross product under coordinates rotation
Question goes as
If $\vec A$ and $\vec B$ are invariant under rotation, the prove that $ \vec A \times \vec B $ is also invariant.
However solution of on the other page is not given. Says ...
5
votes
3answers
92 views
How to divide by $(a_1,a_2,a_3)$
I have been searching for an explanation in Howard's Linear Algebra and couldn't find an identical example to the one below.
The example tells me that vectors $\boldsymbol{a}_1$, $\boldsymbol{a}_2$ ...
1
vote
3answers
278 views
Cross product and dot product
What's the easiest way to understand and prove that $A \cdot B \times C = C \cdot A \times B $ ?
2
votes
1answer
84 views
Vector question, solving $r\wedge a=b$ and $r\wedge c=d$, with conditions
I am stuck on the following
Show that the vector equation $r\wedge a=b$ has a solution $$r=\lambda a + \frac {a \wedge b}{|a|^{2}}$$
Show that the vector $r\wedge a=b$ and $r\wedge c=d$, with ...
2
votes
2answers
150 views
Vectors question
I'm trying to prove whether the followings statements are true or not. I would appreciate your help, as I'm not sure how to begin.
Given:
$ u,x_n \in \mathbb{R}^3$ and for every $n$, let $x_{n+1}=u ...
1
vote
1answer
2k views
Fleming's “right-hand rule” and cross-product of two vectors
I have been throwing around hand gestures for the past hour in a feeble attempt at trying to solve this question involving a cross product of two vectors $a$ x $b$. So far, I haven't found any ...
1
vote
1answer
708 views
Cross product in complex vector spaces
When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors?
I suppose that one possible ...
4
votes
6answers
746 views
Visual Ways to Remember Cross products of Unit vectors? Cross-product in $\mathbb F^3$?
Objective to find visual and accessible ways to remember this formula fast
$$(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)$$
I have used Sarrus' rule but it is slow, more here. Since it is slow, I have ...
