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No. A vector is actually a first order tensor as opposed to all tensors being vectors (vector quantities could be considered a subset of the set of all tensor quantities) because if you were to take a vector in three spatial dimensions A it can be defined by the equation A=A1e1+A2e2+A3e3 and also follows the tensor transformation laws given by A'i=αi'kAk for instance. Tensors however are actually more generalised objects which include vectors, scalars (zeroth order tensors) and more complicated systems.
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