Mar 22, 2019 · 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...

$\begingroup$ It is true, 2 vectors can only yield a unique cross product in 3 dimensions. However, you can yield a cross product between 3 vectors in 4 dimensions. You see, in 2 dimensions, you only need one vector to yield a cross product (which is in this case referred to as the perpendicular operator.). It’s often represented by $ a^⊥ $.

May 25, 2012 · In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?

Feb 15, 2016 · The difference between an ordered pair of vectors and a tensor product of vectors is that if you multiply one of the vectors by a nonzero scalar and the other by the reciprocal of that scalar, then you get a different ordered pair but the same tensor product. $\qquad$ $\endgroup$ –

Thus, the cross product of two unit vectors $\vec{u}$ and $\vec{v}$ is itself a unit vector if and only if $\vec{u}$ and $\vec{v}$ are orthogonal, i.e. meet at right angles (this makes $\sin(\theta)=\sin(\frac{\pi}{2})=1$). As to the general interpretation of the magnitude of the cross product, see Wikipedia:

Jul 25, 2017 · $\begingroup$ A search for "generalized cross product" turns up a number of questions likely to be of interest, including Is the vector cross product only defined for 3D? and Generalized Cross Product. ;) (Not marking as a duplicate because you're better able to judge which question, if any, most nearly matches yours.) $\endgroup$ –

Dec 28, 2023 · 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 ...

Sep 8, 2016 · The complex cross product is commonly used in physics to compute torques (force $\times$ lever arm), and also the intensity of fields in electricity and magnetism (remember the right-hand rule?). Given complex numbers a and b, where v,w,x,y are real numbers and i is the imaginary number:

Oct 12, 2015 · You can't expect to do linear algebra with a curvilinear coordinate system. E.g. using this determinant, a simple cross product of the x and y unit vectors would give an r of pi^2 / 4 instead of 1. $\endgroup$ –

Then there is the cross product (vector), which is used when we only care about perpendicular components of the vectors — for example when calculating torque on a door being opened $\vec{\boldsymbol \tau} = \mathbf{\vec F} \times \mathbf{\vec r}$, we only care about the component of the force applied perpendicular to the door. As a bonus, the ...