Induced voltage of a conductor in a magnetic field 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?
 A: 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.
A: Work is scalar; why does it have a direction?
Voltage is not a vector; however, when we talk about the direction of voltage, we mean to the change in potential across the voltage source. The change can be positive or negative.
To make things clear, we will see an example to understand what the sign of voltage really means.


Maxwell-Faraday equation dictates that the E.M.F induced across the inductor will be in a direction such that it tries to reduce the change in current. Mathematically, we can show it as:
$$ \mathcal{E}_{induced} = -L\frac{di}{dt}$$
The current in circuit starts from the positive terminal (high potential) of the battery and travels through the wire to reach the negative terminal (low potential) of the battery. It is not a surprise; current flows from higher potential to lower potential. If you are measuring the voltage across the battery from the negative to the positive terminal, it'll be $+ve$ because the potential increased. Afterall, voltage is the potential difference. 
Unlike a battery, an inductor is going to be nasty. It will try its level best to prevent any change (increasing in this particular example) in the current. It certainly cannot stop the current but it can try to delay or slow down the change in current. For this, the induced E.M.F appears to work like a battery in reverse. If you measure the potentials at the two ends of the inductor, you'll find that the potential at the left end is higher than the potential at the right end. The potential decreased. Therefore, the potential difference (or E.M.F or voltage across the inductor) is negative. This is what the negative sign for the induced voltage means. You can use the same idea to explain why the potential difference across the resistor is negative as well.
