In this link: https://math.wikia.org/wiki/Blade_(geometry)
It states that blades are a generalization of vectors. Fair enough.
But then in this link: https://math.wikia.org/wiki/Tensor
It states that tensors are a generalization of vectors.
Now that has me confused. It doesn't seem like blades and tensors are the same thing, but how can that be if they both generalizations of the same thing? Unless there is more than one way to generalize the same thing? Though, if possible, that in itself seems strange also.
Are blades a geometric generalization of vectors? While tensors are a matrix (algebraic?) generalization of vectors? But even if that's the case, that seems weird too since it seems like the definitions would quickly converge so having two different names for the same thing seems pointless.
I never heard the word "blade" but from the definition it looks like blades are a particular case of tensors.
More precisely, a blade is the tensor given by the skew-symmetric product of a set of vectors of a vector space $V$:
$$ v_1 \wedge \cdots \wedge v_k = \sum_{\sigma \in \mathfrak{S}_k} (-1)^\sigma v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(k)} $$ where the sum is over the full permutation group on $k$ elements and $(-1)^\sigma$ is the sign of the permutation $\sigma$.
In this context, blades can also be interpreted as elements of the (affine cone over the) Grassmannian of $k$-planes in its Plucker embedding in $\Lambda^k V$.