calculus - Derivative of cross-product of two vectors
2012年5月25日 · You can evaluate this expression in two ways: 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: d dt(u × v) = du dt × v + u × dv dt. Picking a method depends on the problem at hand. For example, the product rule is used to derive Frenet Serret formulas.
What's the cross product in 2 dimensions? [duplicate]
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...
Connection between cross product and determinant
2023年12月28日 · 2. If →i, →j, →k are the three basic vectors of R3 then the cross product of vectors (a, b, c), (p, q, r) is the determinant of the matrix (→i →j →k a b c p q r) by definition. The coordinates of that vector are obtained by expanding this determinant along the first row. Share.
Why is cross product only defined in 3 and 7 dimensions?
But I would like to remark that as quoted in the wikipedia article you linked, there are 480 different cross products in $\mathbb{R}^7$. This suggests to me that it is not a very natural operation - at least if interpreted as a cross product. The natural extension of the cross product to higher dimensions is what is called wedge product ...
How to prove the distributive property of cross product
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 …
divergence of the cross product of two vectors proof
2016年1月11日 · Divergence of cross product of two vectors, intuition. Related. 2. Distance between line and point in ...
linear algebra - Cross product in complex vector spaces
2015年3月13日 · $\begingroup$ The meaning of triple product (x × y)⋅ z of Euclidean 3-vectors is the volume form (SL(3, ℝ) invariant), that gets an expression through dot product (O(3) invariant) and cross product (SO(3) invariant, a subgroup of SL(3, ℝ)). We can complexify all the stuff (resulting in SO(3, ℂ)-invariant vector calculus), although we ...
Is the cross product of two unit vectors itself a unit vector?
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:
Cross product- square - Mathematics Stack Exchange
2015年4月17日 · First of all, the cross product is associative, i.e. (→u + →v) × →w = (→u × →w) + (→v × →w). The same is true on the left-hand side too. Moreover, the cross product is linear, i.e. (λ→u) × (μ→v) = λμ(→u × →v). Finally, recall that for any vector →u we have →u × →u = →0. We can apply these rules to ...
linear algebra - Why does cross product tell us about clockwise or …
The cross product is $$\eqalign{ {\bf A}\times {\bf B}=\left|\matrix{{\bf i}&{\bf j}&{\bf k}\cr a&0&0\cr b&c&0\cr}\right| &=\langle 0,0,ac\rangle.\cr}$$ As predicted, this is a vector pointing up or down, depending on the sign of ac. Suppose that a>0, so the sign depends only on c: if c>0, ac>0 and the vector points up; if c< 0, the vector ...