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import torch
def normalize(v: torch.Tensor) -> torch.Tensor:
return v / torch.linalg.norm(v, dim=-1, keepdim=True)
def cross_product(v1: torch.Tensor, v2: torch.Tensor) -> torch.Tensor:
return torch.stack(
[
v1[..., 1] * v2[..., 2] - v2[..., 1] * v1[..., 2],
-(v1[..., 0] * v2[..., 2] - v2[..., 0] * v1[..., 2]),
v1[..., 0] * v2[..., 1] - v2[..., 0] * v1[..., 1],
],
dim=-1,
)