# Tagged Questions

In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.

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### 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 ...
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### 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 ...
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### 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 $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\boldsymbol{x}_1=t\boldsymbol{v}$ (...
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### 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: ...
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### 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 ...
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### 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 ...
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### Higher dimensional cross product

I know that cross products do not exist in 4, 5 or 6 dimensions, but do in 7 dimensions. So I was wondering if this was because cross products can be considered the imaginary part of $2^n - ion$ ...
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### cross-products versus units of measure

If I draw 2 perpendicular line segments on the ground, 3 meters and 4 meters, how far into the sky does their cross-product extend? What if instead the line lengths are 300 cm and 400 cm? Can ...
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### 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.
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### 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 ...
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### 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 ...
By matrix-defined, I mean $$\left<a,b,c\right>\times\left<d,e,f\right> = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|... 2answers 331 views ### How to prove the equality of two vectors? OK, i am trying to prove that if \vec a\times \vec b = \vec a \times \vec c and also \vec a\cdot \vec b = \vec a \cdot \vec c then \vec b = \vec c. so far i got to \vec n \tan \alpha = \vec m ... 1answer 3k views ### How do you integrate Cross Products? Hey I'm doing a course in mechanics and these keep cropping up! So for this question I'm working in 3d, and so far have$$m \mathbf{k} \cdot (\mathbf{q} \times \ddot{\mathbf{q}} )=0$$so I need ... 2answers 76 views ### Ratio of area formed by transformed and original sides of a parallelogram I am interested in finding the ratio of area formed by transformed and original sides of a parallelogram, given by:$$\frac{\|Ma\times Mb\| }{\| a\times b \|}$$M is a 3 \times 3 matrix and  a, b... 2answers 326 views ### Line integrals, cross products, surface integrals and Stoke's Theorem related problem? The vector field \vec{F}(\vec{R}) is defined as being equal to the line integral over some simple closed curve C:$$\vec{F}(\vec{R})=\oint_C\|\vec{r}-\vec{R}\|^2d\vec{r}.$$We show that there ... 2answers 81 views ### Using Gram-Schmidt to compute the cross product of 3 vectors in \Bbb R^4 [duplicate] I want to ask about vector multiplication (cross product) in 4-d. I heard that Gram-Schmidt process is involved but I am not sure how the process is involved. The multiplication involves 3 ... 1answer 483 views ### Determine Cross Product with Left Hand vs Right Hand If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ... 6answers 197 views ### Unit vectors that are orthogonal to vectors I have to find all the unit vectors that are orthogonal to the vectors \overrightarrow{a}=(2, -4, 3), \overrightarrow{b}=(-4, 8, -6) . I calculated that the cross product \overrightarrow{a} \... 3answers 4k views ### How to prove this vector identity [closed] How do i prove this vector identity ?$$(\vec a \times \vec b)\times \vec c=(\vec a \cdot\vec c)\vec b - (\vec b\cdot\vec c)\vec a
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 ...