Is this a group?

Algebra Level 3

For vectors \((x_{1}, y_{1}, z_{1})\) and \((x_{2}, y_{2}, z_{2})\) in \(\mathbb{R}^{3}\), define the cross product \(\times\) as such:

\[(x_{1}, y_{1}, z_{1}) \times (x_{2}, y_{2}, z_{2}) = (y_{1}z_{2}-z_{1}y_{2}, z_{1}x_{2}-x_{1}z_{2}, x_{1}y_{2}-y_{1}x_{2})\]

Does \(\mathbb{R}^{3}\) form a group under the cross product?

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