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A book which I referenced for Electrical Machinery states that the voltage induced in a conductor inside a magnetic field is given by

$$ \mathcal{E}=(\mathbf{v} \times \mathbf{B})\cdot \mathbf{l}$$

Since all three are vectors, and the cross multiplication inside the brackets results in another vector and that value is being dot product with another vector, the last result should be a scalar. But how can the last result (induced voltage) have a direction then?

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thanx.. that explains it :) – DesirePRG Jun 5 '13 at 4:27
@userØØ7 either \mathbf or \vec. Not both. – dmckee Jun 5 '13 at 4:45
Voltage does not have direction. It has a sign which corresponds to two possible ways the potential difference may be set up. – Ján Lalinský Nov 2 '15 at 16:06

Voltage does not have direction.... It's just high potential or low potential.The direction of vector $\vec v \times \vec B$ points in direction of high potential .

Also if you see the $\mathbb E=\vec v\times \vec B. \vec l$ . It really should not depend on what direction $\vec l$ is pointing, so we find the high potential end physically. So, always take magnitude of the $\mathbb E$ you get and see direction physically to avoid problems.Otherwise sometimes working in vectors can give -ve results which can create confusion.

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