It depends on the type of product used.
A dot or scalar product of two vectors will result in a scalar.
A cross or vector product of two vectors will result in a vector.
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Charge is not a vector.
Because there are two different ways of computing the product of two vectors, one of which yields a scalar quantity while the other yields a vector quantity.This isn't a "sometimes" thing: the dot product of two vectors is always scalar, while the cross product of two vectors is always a vector.
It is the cross product of two vectors. The cross product of two vectors is always a pseudo-vector. This is related to the fact that A x B is not the same as B x A: in the case of the cross product, A x B = - (B x A).
Product
The product of two vectors can be done in two different ways. The result of one way is another vector. The result of the other way is a scalar ... that's why that method is called the "scalar product". The way it's done is (magnitude of one vector) times (magnitude of the other vector) times (cosine of the angle between them).