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 Why the direction of dipole moment is from negative to positive charge? 

The torque on a dipole is given by,

The torque obtained is positive when the direction of the dipole is taken from negative charge to positive charge.

Another reason is, in analogy with magnetic dipole where field outside the magnet is from North to South while field within the magnet is from South to North, the direction of electric dipole within the dipole is taken from negative charge to positive charge (this is just an analogy that is followed, in practical direction of electric lines of force is always from positive to negative charge).

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Resultant vector: Its best to draw a little diagram (no math involved...just a sketch)

Take a positive charge and put it near a negative charge (dipole). Draw all of the lines pointing away from the positive ...and a the negative lines pointing towards the negative. Notice what happens in between....they are going the convention way (positive to negative). But, what happens on the outsides if you ignore the direction in between. Thats right......start near the negative charge and follow the field lines (on the outside) and draw a long arrow (ignoring the in between fields) to the outside of the positive charge. Notice what direction its in.......its from negative to positive.

That is whats happening. In a dipole the charges are so close together that the major component to the field at a large distance away from dipole is from the outside. The in between fields become negligible

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electric dipole moment is a measure of the seperation of positivr and negative charges in a system of charges ie., it is a measurde of the system's overall polarity.
hence P = q L, where L is the displacement vector pointing from negative charge to the positive charge.Thus P points from negative charge to positive charge.There is no inconsistency here, because P has to do with orientation of the dipole ie., the position of charges and does not indicate the direction of the field orginating in these charges.electric dipole moment is a measure of the seperation of positivr and negative charges in a system of charges ie., it is a measurde of the system's overall polarity.
hence P = q L, where L is the displacement vector pointing from negative charge to the positive charge.Thus P points from negative charge to positive charge.There is no inconsistency here, because P has to do with orientation of the dipole ie., the position of charges and does not indicate the direction of the field orginating in these charges.

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