For questions on cross products.
12
votes
4answers
1k 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 ...
10
votes
4answers
2k views
Wedge product and cross product - any difference?
I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
9
votes
4answers
650 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 ...
9
votes
1answer
635 views
Wedge Product, A Novel Interpretation or Just Plain Wrong?
I have read (I think) all of the previous threads on this website (and many others)
on this topic & unfortunately have not found an answer to my question. Due to the fact that I am only beginning ...
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 ...
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 ...
5
votes
2answers
926 views
What's the opposite of a cross product?
For example, $a \times b = c$
If you only know $a$ and $c$, what method can you use to find $b$?
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 $\boldsymbol{a}_1$, $\boldsymbol{a}_2$ ...
5
votes
1answer
122 views
The proof of $\hat{b}(\hat{a}\cdot\hat{c})-\hat{c}(\hat{a}\cdot\hat{b})=\hat{a}\times(\hat{b}\times\hat{c})$
formula: $\hat{b}(\hat{a}\cdot\hat{c})-\hat{c}(\hat{a}\cdot\hat{b})=\hat{a}\times(\hat{b}\times\hat{c})$
$\hat{a}\times(\hat{b}\times\hat{c})$ is on the $\hat{b}$, $\hat{c}$ plane, so:
...
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 $R^T R = I$ )
...
4
votes
6answers
741 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 ...
4
votes
1answer
391 views
Cross product of vectors as a determinant: valid matrix operation?
"The definition of the cross product can also be represented by the
determinant of a formal matrix."
—Wikipedia
This seems like a hack to me—something of much practical use but ...
4
votes
1answer
92 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
39 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:
$$
\begin{pmatrix}
R \\
G \\
B
...
3
votes
2answers
267 views
Example of an associative cross product, any significance?
While trying to find cases that showed the cross product is not associative, I found some that were. I'm trying to show that
$(\mathbf{A}\times \mathbf{B}) \times \mathbf{C} \ne \mathbf{A}\times ...
3
votes
5answers
395 views
Help understanding cross-product
I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
3
votes
2answers
122 views
Why is $\det(\vec{A},\vec{B}) = |\vec{A} \times \vec{B}|$?
In the multivariable calculus class the teacher showed us the formula of the cross product
$$ \vec{A} \times \vec{B} =\begin{vmatrix}\hat{\imath}& \hat{\jmath}& \hat{k} \\
a_1 & a_2 ...
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.
2
votes
2answers
126 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 ...
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
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 ...
2
votes
2answers
53 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 $\bf{y}$ ...
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 ...
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 ...
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
1answer
322 views
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
2
votes
1answer
638 views
How do you integrate Cross Products?
Hey I'm doing a course in mechanics and these keep cropping up!
So for this question I'm working in 3d, and so far have
$$m \mathbf{k} \cdot (\mathbf{q} \times \ddot{\mathbf{q}} )=0$$
so I need ...
2
votes
3answers
33 views
Cross product- square
I recently saw the following expression somewhere-
$$\frac{1}{2} \left\| \frac{\vec{u}}{9} \times \frac{\vec{u} + \vec{v}}{9} \right\| + \frac{1}{2} \left\| \frac{\vec{u} + \vec{v}}{9}\times ...
2
votes
0answers
112 views
Cross product in higher than 3 dimensions
As I understand it, to get an $n$-dimensional cross product, you need $n-1$ vectors of dimension $n$. However my lecture notes are quite miss leading in the fact that they suggest this isn't always ...
2
votes
1answer
233 views
Determining partial derivatives and cross products for bicubic interpolation using function values only?
I'm trying to implement a bicubic interpolation algorithm. In order to calculate the interpolated values, I need to calculate sixteen coefficients used in the calculation process - and that's where ...
2
votes
1answer
221 views
Minimize sum of the norm of cross products
Here I have an interesting problem on linear algebra. It looks very simple, but not so easy to solve for me.
Let $r_i, i=1,…,n$ be unit vectors in $R^n$, find a unit vector $x$ to minimize
$\sum \| ...
1
vote
3answers
233 views
Area of a parallelogram (linear algebra)
Find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.
1
vote
2answers
35 views
Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is maximal, and then a unit vector $w$ s.t $|(u\times v)\cdot w|$ is minimal
This is a similar question to the one I have posted before. The problem
is as in the title:
Given $u=(-2,5,3)$ find a unit vector $v$ s.t $|u\times v|$ is
maximal, and then a unit vector $w$ s.t ...
1
vote
1answer
117 views
proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$
I encounter this triple product property in wikipedia
But I can't find proof for
$$[\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
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?
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
70 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
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 ...
1
vote
3answers
277 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 $ ?
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
29 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
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
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 $|a|=|b|=|c|=1$
1
vote
2answers
129 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 ...
1
vote
1answer
705 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 ...
1
vote
1answer
164 views
Deduce plus and minus with Cross Product in 3th and 4th Maxwell equations
The laws:
$\nabla \times \bar{E} = \bar{I}_{m} - \frac{\partial \bar{B}}{\partial \bar{t}}$
$\nabla \times \bar{H} = \bar{J}_{f} + \frac{\partial \bar{D}}{\partial \bar{t}}$
so how can I remember ...
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
77 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, ...
1
vote
1answer
105 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 ...
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.
...
