For questions on cross products.

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3
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2answers
1k 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
157 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
68 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: ...
3
votes
1answer
460 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
74 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 ...
22
votes
6answers
6k 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
62 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
923 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
97 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
600 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
114 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
158 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
6k 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
2k 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 ...
5
votes
6answers
4k 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 ...
2
votes
1answer
2k 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 ...
0
votes
1answer
220 views

Simplify $A \times (A \times B)$

Where $A$ and $B$ are vectors, and $\times$ is the cross product operator. I was able to get $A(A \cdot B) - B$ using the vector triple product, but it doesn't look like a simplified version to me.
1
vote
1answer
1k views

Cross product of partial derivatives in surface integrals

I need help in understanding how to compute the cross product of two partial derivatives to help me calculate a surface area. I've watched the Khan Academy lecture on the subject but they seem to be ...
5
votes
1answer
141 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
2answers
626 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 ...
6
votes
1answer
900 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
2answers
138 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 ...
18
votes
4answers
4k 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 ...
3
votes
5answers
543 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. ...
2
votes
1answer
661 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
1
vote
1answer
213 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 ...
0
votes
1answer
1k views

normalized cross product

Is there a way to get the result of a cross product to be normalized after just a cross action, i.e. without doing after the cross v/|v|? (the vectors involved are ...
8
votes
3answers
3k 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 $b$?
14
votes
1answer
1k 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 ...
2
votes
1answer
321 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 $\mathbb{R}^n$, find a unit vector $x$ to minimize ...
12
votes
4answers
5k 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) ...