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Magnetic Susceptibility, denoted as χ is defined as a scalar, represents the ratio between the Magnetization vector ($\vec {M}$) and the Magnetic field strength vector ($\vec{H}$).

However, in mathematics, division of two vectors is not defined. How can we then define Magnetic Susceptibility as the ratio of two vectors? Should we consider the ratio of the x-components of M and H as $\chi_x$ or something else?

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The magnetic susceptibility is generally a tensor quantity (e.g. a matrix). That is the most general way to linearly transform one vector into another. When things are isotropic, the susceptibility is proportional to the identity matrix as $$\mathbf{\chi}=\chi_0 \mathbf{I}$$ and you can treat it as a scalar quantity.

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