Vectors

Cross Product is only valid in R^3 or R^7. I am interested in knowing why it's not valid in 2D space.I tried to invoke it out while solving a proving problem and in the problem I took two 2D vectors and when did a cross over them got all thing along +Z axis x X y = z so how to get a physical sense or an intuition behind that (cross product invalid in 2D).

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

The structure of vector products in $\mathbb{V}^3$ (three-dimensional vector space over an arbitrary field), which correspond to the usual Euclidean vector product and any generalisation based on arbitrary symmetric bilinear forms, is very different to that in $\mathbb{V}^7$ , where certain luxuries that are present in our original framework do not hold, e.g. the Jacobi identity and Lagrange's formula.

In fact, the idea of a vector product outside of $\mathbb{V}^3$ should not be tackled within the realm of linear algebra. Thus, we must look outwards and rely on tools from geometric algebra. The notion of exterior products being a subspace of vectors in $\mathbb{V}^n$ perpendicular to a number of vectors is easier to interpret geometrically and algebraically, and is an ideal tool for extending vector products outside of $\mathbb{V}^3$ .

To answer your question directly, look up Hurwitz's theorem. Again, working outside linear algebra and extending to geometric algebra is much more intuitive and less taxing on the current framework.

A Former Brilliant Member - 2 years, 9 months ago

Help with this https://brilliant.org/discussions/thread/vector-analysis/

Rishi Tiwari - 2 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$