Linked Questions

26
votes
2answers
13k views

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 ...
0
votes
2answers
48 views

What's the cross product in 2 dimensions? [duplicate]

The math book i'm using states that the cross product for two vectors is defined over $R^3$: $$u = (a,b,c)$$ $$v = (d,e,f)$$ is: $$u \times v = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\...
2
votes
0answers
76 views

What dimensions permit a cross product [duplicate]

A cross product is possible in a $3$D and in a $7$D system. What prevents a cross product from being possible in a vector system of higher number of dimensions? For instance $15$D or $2^n$$-1$ ...
2
votes
0answers
44 views

Cross product in 4 dimensions [duplicate]

Cross product is defined in three dimensions the resulting vector have the same magnitude as the area of the parallelogram formed by the 2 multiplied vectors and its direction direction is ...
139
votes
15answers
18k views

What's new in higher dimensions?

This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in ...
28
votes
4answers
6k views

Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have ...
12
votes
2answers
36k views

Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = \...
18
votes
1answer
3k views

Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ ...
4
votes
3answers
9k views

Finding a fourth vector that makes a set a basis

The following vectors are linearly independent - $v1 = (1, 2, 0, 2)$ $v2 = (1,1,1,0)$ $v3 = (2,0,1,3)$ Find a fourth vector v4 so that the set { v1, v2, v3, v4 } is a basis fpr $\mathbb{R}^4$? I ...
0
votes
2answers
769 views

Proving an orthogonal basis of vectors in ℝ4

For an orthogonal set of vectors {$\vec{v1}$,$\vec{v2}$,$\vec{v3}$} in $\mathbb{R}^{4}$, show that there is a vector $\vec{v4}$ so that {$\vec{v1}$,$\vec{v2}$,$\vec{v3}$,$\...
3
votes
1answer
432 views

Why the cross product between 2 vectors exists only in 3 dimensions?

Why the cross product between 2 (two!) vectors exists only in 3 dimensions? The article "Cross Products of Vectors in Higher Dimensional Euclidean Spaces" by W. M. Massey (see http://doi.org/10.2307/...
3
votes
1answer
130 views

Impossibility of nontrivial product of vectors

When introducing the tensor product of vectors, Serge Winitzki in his work Linear Algebra via Exterior Products claims that It turns out to be impossible to define a nontrivial product of vectors ...
0
votes
0answers
65 views

Cross product between two n-dimensional vector $(n = 4-5)$

As we know, we can write the cross product between two three dimensional vectors as a matrix-vector product. Let two vectors are $V = \left[ \begin{array}{c} V_1 \\ V_2 \\ V_3 \end{array} \right] \...