For questions on cross products.

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26
votes
6answers
8k 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 ...
5
votes
4answers
14k 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 ...
5
votes
6answers
5k 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 ...
12
votes
2answers
2k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
23
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
576 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
3answers
11k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that ...
8
votes
3answers
4k 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 $b$?
6
votes
1answer
832 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 ...
11
votes
5answers
5k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
2
votes
3answers
632 views

showing / proving curl identity $\nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$

OK, I have to show the following: $$ \nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$$ This should be pretty easy, but I wanted to be sure I was doing this correctly. I set up the matrix: ...
4
votes
0answers
207 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
1
vote
3answers
720 views

How come the cross product of two planes is collinear with the direction vector of the line?

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.
0
votes
1answer
167 views

Category-theoretic cross product and set-theoretic cross product

I recently proved as an exercise the associativity of cross product as defined in category theory. But in set theory, cross product is not associative. It seems intuitive to me that cross should be ...
0
votes
1answer
397 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 ...
2
votes
5answers
126 views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
2
votes
2answers
255 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
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 ...
1
vote
2answers
191 views

Line integrals, cross products, surface integrals and Stoke's Theorem related problem?

The vector field $\vec{F}(\vec{R})$ is defined as being equal to the line integral over some simple closed curve $C$: $$\vec{F}(\vec{R})=\oint_C\|\vec{r}-\vec{R}\|^2d\vec{r}.$$ We show that there ...
0
votes
1answer
58 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
2k views

How to prove this vector identity

How do i prove this vector identity ? $$(\vec a \times \vec b)\times \vec c=(\vec a \cdot\vec c)\vec b - (\vec b\cdot\vec c)\vec a$$
0
votes
2answers
73 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 ...