2012年5月25日 · You can find the cross product first, and then differentiate it. Or you can use the product rule, which works just fine with the cross product: $$ \frac{d}{dt}(\mathbf{u} \times \mathbf{v}) = \frac{d\mathbf{u}}{dt} \times \mathbf{v} + \mathbf{u} \times \frac{d\mathbf{v}}{dt} $$ Picking a method depends on the problem at hand.

2019年3月22日 · E. A. Abbott describes a 2D cross product nicely in his mathematical fantasy book "Flatland": Flatland describes life and customs of people in a 2-D world: in this universe vectors can be summed together and projected, areas are calculated, rotations are clock-wise or counter clock-wise, reflection is possible...

2023年12月28日 · You might have two vectors whose cross product is $(5, 3, 2)$ under regular coordinates, but if you changed your coordinate system to switch the first and second dimensions, without (anti)symmetry the cross product could have an entirely different value, like $(-1, 4, 1)$. A mathematical operation that depends on something totally unphysical ...

2015年4月17日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2013年4月15日 · The geometric definition of the cross product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is given by $$\mathbf{A} \times \mathbf{B} \overset{\operatorname{def}}{=} |\mathbf{A}||\mathbf{B}|\sin \theta \,\hat{\mathbf{N}},$$ where $\theta \in [0,\pi]$ is the angle between the two vectors when placed tail to tail, and $\hat{\mathbf{N}}$ is a unit vector whose …

Cross product of two vectors, given magnitudes and angle. Ask Question Asked 10 years, 2 months ago. ...

2015年3月16日 · That is the magnetic field. And since the rate of rotation is given by the cross product of the rotation vector by the vector, the rate of change of the velocity $\vec v$ of a test charge at the origin is given by $\vec B\times \vec v$. Now let me attempt to answer OP's questions. Why wasn't the cross-product defined as just this magnitude?

2020年5月26日 · is an entirely different vector (cross-products are not associative). All higher vector products can be similarly reduced, often by repeated application of Eq. 1.17, so it is never necessary for an expression to contain more than one cross product in any term. For instance,

The composition algebra-cross product correspondence Now, given any composition algebra $\Bbb A$ (perhaps assuming $\text{char } F \neq 2$), we can build a cross product on a codimension-$1$ subspace of $\Bbb A$: Let $\bar{\cdot} : \Bbb A \to \Bbb A$ be the linear map that restricts to the identity on $\langle e \rangle$ and the negative of the ...

2016年2月15日 · If you can come up with a matrix such that if you multiply a vector by, the result is the cross product of the vector, then that acts as sufficient proof that it is a linear transformation. $\endgroup$