# Tag Info

14

A little bit more of the 'how and why': the dot product comes about as a natural answer to the question: 'what functions do we have that take two vectors and produce a number?' Keep in mind that we have a natural function (addition) that takes two vectors and produces another vector, and another natural function (scalar multiplication) that takes a vector ...

12

I would have left this as a comment, but apparently I can't comment yet. To answer your final question, why introduce the wedge, the point is that the wedge product (as explained in the Wikipedia article) is a notion that generalizes to R^n and indeed any vector space---in general the output is what is called a "bivector". Now, it just so happens that in ...

11

Yes, you are correct. You can generalize the cross product to $n$ dimensions by saying it is an operation which takes in $n-1$ vectors and produces a vector that is perpendicular to each one. This can be easily defined using the exterior algebra and Hodge star operator http://en.wikipedia.org/wiki/Hodge_dual: the cross product of $v_1,\ldots,v_{n-1}$ is ...

8

A longuish comment The cross product satisfies the Jacobi identity $$(a\times b)\times c+(b\times c)\times a+(c\times a)\times b=0.$$ Using this and the fact that it is antisymmetric, you can easily see that $$(a\times b)\times c=a\times(b\times c)\iff(c\times a)\times b=0.$$ This immediately explains your example where the equality holds and the one where ...

7

No difference at all. I've been trying to write a little proof, but the software on this page seems to have forgotten how to write maths. :-( Anyway: I assume that by "regular cross/vector product" you mean the definition with coordinates as in Wikipedia. Try to compute both sides of your equation $(u\wedge v ) \cdot w = \det (u, v, w)$ with your ...

7

This is most easily proved without coordinates. The cross product is the unique vector that is orthogonal to both factors, has length given by the area of the parallelogram they form and forms a right-handed triple with them. These properties are all invariant under rotations, and thus so is the cross product.

7

Hint: the area of a parallelogram (see left-most image) is equal to the determinant of the $2\times 2$ matrix formed by the column vectors representing component vectors determined by the given points. $$A = \text{det}\,\left(\vec u \;\; \vec v\right)$$ The area of a parallelogram is also equal to the magnitutude of the cross product of the component ...

7

No, it's not an accident. The cross product is orthogonal to each factor, so the vector has to be orthogonal to $b\times c$, hence in the plane spanned by $b$ and $c$. But it also has to be orthogonal to $a$. So, writing $$a\times(b\times c) = xb + yc$$ and dotting with $a$, you get $x(b\cdot a) + y(c\cdot a)=0$. So the answer must be some scalar multiple of ...

6

The name "product" for the cross product is unfortunate. It really should not be thought of as a product in the ordinary sense; for example, it is not even associative. Thus one should not expect it to have properties analogous to the properties of ordinary multiplication. What the cross product really is is a Lie bracket.

5

The best introduction I know of to the exterior product is Sergei Winitzki's free book Linear Algebra via Exterior Products. Chapter $2$ in particular I think addresses all of your questions (it is unclear how much of Chapter $1$ you need to read in order to read Chapter $2$, I guess that depends on how much linear algebra you've had).

4

The Cartesian product of sets is what is usually referred to as the product of two sets, and it is denoted with $\times$. It's certainly associative. The intuition is that when you form $(X\times Y)\times Z$ and $X\times (Y\times Z)$, they are both basically the same as the set of ordered triples $X\times Y\times Z$. While the first two are not equal as ...

4

It's how the cross product is defined. Operations are generally things we define to have some nice properties. Even the notion of addition becomes non-intuitive and somewhat unnatural when you start involving irrational numbers. As far as anti-symmetry, this is why we use the "right hand rule." Align your right hand along the first vector, curl your fingers ...

4

All quantities below are vectors. I will use the following properties of cross-products and dot-products: $$(x \times y) \times z = (x \cdot z) y - (y \cdot z)x \\ x \cdot ( y \times z) = y \cdot (z \times x) = z \cdot (x \times y) \\ x \cdot (x \times y) = 0$$ We start with the righthand side. For convenience, denote $a \times c = v$. Then ...

4

Here are a few references about the history of linear algebra: "A Brief History of Linear Algebra and Matrix Theory" "History of Linear Algebra" Cross Product - Wikipedia: History The Wikipedia article seems to address your question most directly.

4

I'm not exactly sure what you are asking, however the following may be useful as a less verbose way of obtaining the same result. The key fact is that the cross product of $A,B$ is the unique element $A\times B$ such that $\langle x, A\times B \rangle = \det \begin{bmatrix} A & B & x\end{bmatrix}$, $\forall x$. Let $Q$ be a rotation (ie, $Q^TQ = ... 3 Let$C = \omega_1$and$<$be the ordinal ordering of$C$. Since$C = \omega_1$, it is the least uncountable ordinal (cardinal). Hence for all$\eta \in \omega_1$,$\{\alpha < \eta\}$is countable. Define$A \subset C^2$as follows:$A = \{(\alpha,\beta) \in C^2 : \alpha < \beta\}$. Then for any fixed$\alpha$,$\{\beta : (\alpha, \beta) \in ...

3

Assign $u$ to be the unit normal vector to some plane $S$ in $\mathbb{R}^3$, then $u\times x_n \in T(S)$ for any $n$, which means $u\times x_n$ is a vector field on the plane $S$. After the first cross product, all $x_n$'s ($n > 1$) lie on the plane $S$, because $x_{n}\cdot u = (u\times x_{n-1})\cdot u = 0$ for all $n> 1$. Now $x_{n+2} = ... 3 I have figured out that$(\boldsymbol a_2 \times \boldsymbol a_3)$is a vector product which i can calculate like this: $$\boldsymbol a_2 \times \boldsymbol a_3 = \left| \begin{array}{ccc} \boldsymbol{\hat{i}}&\boldsymbol{\hat{j}}&\boldsymbol{\hat{k}}\\ 0&a&0\\ 0&0&a \end{array} \right| =\boldsymbol{\hat{i}} a a + ... 3 I usually teach this (which requires no additional writing and avoids cyclic permutations, which are often confusing for students):$$ \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \times \left[\begin{array}{c} u \\ v \\ w \end{array}\right] $$1) ignore x and u (that is: mentally block view of the first row), compute 2\times 2 ... 3 You're supposed to curl your hand like so: \hskip 1.3in Your knuckles are supposed to form an angle while your fingers are otherwise straight, and your knuckles themselves point in the direction of a while your fingertips point in the direction of b. Using this, part (b) presumably wants you to use RHR to figure out which of the 8 octants \vec{n} ... 3 I really truly believe it's hard to beat the 3\times 3 determinant mnemonic. However, there is a trick for 3 \times 3 determinants which make computing them a snap. Remember that you compute a 2 \times 2 determinant by multiplying diagonals and off-diagonals (the upward diagonal) and taking the difference. You can do the same for 3 \times ... 3 I think the best way to answer your question is, it's a mnemonic. This mnemonic lets you get your hands on a collection of mathematical objects called "exterior forms". From this perspective, it's not a "hack" but to explain exactly what it is essentially requires discussing dual spaces and things not appropriate for standard multivariable calculus ... 3 Yes, this definition is chosen for a reason, as the unique solution to a pedagogical problem. Do Carmo's definition is awkward and redundant in 3 dimensions, but it is the only one among the usual definitions for the cross product that when generalized to n dimensions (there is a cross-product of n-1 vectors in R^n) is rigorous, visibly basis ... 3 Perhaps understanding the following definition of the cross product would eliminate your confusion: For two vectors a and b in \mathbb{R}^3 the function from \mathbb{R}^3 to \mathbb{R} determined by the rule c \mapsto \det[a, b, c] is a linear form on \mathbb{R}^3, that is, it is a real-valued linear function on \mathbb{R}^3. As such, it can ... 3 Since u_r,u_{\phi},u_{\theta} forms a right handed orthonormal frame of unit vectors the rules for computing vectors at a point p expressed in the frame at p is precisely the same as that for the globally constant Cartesian frame. For example,$$ \vec{V}_1 \cdot \vec{V}_2 = 2(3)+\frac{\pi}{3}\frac{\pi}{6}+\frac{\pi}{4}\frac{\pi}{2} $$More generally, ... 3 Those formulas don't actually always hold. For example if you take the vectors$$a = \begin{pmatrix}1\\0\\0\end{pmatrix} \quad b=\begin{pmatrix}0\\1\\0\end{pmatrix} \quad c=\begin{pmatrix}0\\0\\-1\end{pmatrix}$$Then they are orthogonal, but a \times b = \begin{pmatrix}0\\0\\1\end{pmatrix} =-c also c \times a = -b and b \times c = - a. 3 Here is an explanation which is nearer to linear algebra: In {\mathbb R}^3 we have a volume form$$\epsilon:\quad \bigl({\mathbb R}^3\bigr)^3\to{\mathbb R},\qquad (a,b,c)\mapsto \epsilon(a,b,c)\ ,$$which produces for any three given vectors$a$,$b$,$c$the signed volume of the parallelotope spanned by them. It is linear in all three entries, and when ... 3 Suppose we have two linear functions,$f$and$g$, which agree on all the basis vectors of some space. Then they must agree for every vector on that space, because they are both linear, and a linear function is completely determined by its values on the basis. In gory detail, suppose that we know that$f(\vec{e_i}) = g(\vec{e_i})\$ for each basis vector ...

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