# 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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### 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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### 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 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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### 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) ...
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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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### 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 ...
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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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### 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 $b$?
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### 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 ...
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### 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 $\boldsymbol{a}_1$, $\boldsymbol{a}_2$ ...
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### Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. I ...
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### Can cross products be defined without coordinates?

I recently learned about cross products and understood that cross products can be computed without an origin and coordinates in three dimensions, like vectors can be defined without coordinates. But ...
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### 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 $R^T R = I$ ) I'...
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### Exterior Product vs Cross Product

I was confused about the relationship between a set of basis vectors in 3D, $\left\{\hat e_1, \hat e_2, \hat e_3 \right\}$ and their exterior products. In my head, it makes sense that the identity ...
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### Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
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### 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 ...
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### 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 I'...
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### Octonionic formula for the ternary eight-dimensional cross product

A cross product is a multilinear map $X(v_1,\cdots,v_r)$ on a $d$-dimensional oriented inner product space $V$ for which (i) $\langle X(v_1,\cdots,v_r),w\rangle$ is alternating in $v_1,\cdots,v_r,w$ ...
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### What is the intuition behind the cross product? [duplicate]

I've researched lots of places and still cannot wrap my head around the cross product. The closest thing I have to an understanding of the cross product is that its a measure of how orthogonal two ...
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### Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$\frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b),$$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
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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 ...
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 ...