Thanks for your very valid observations. You are absolutely right by the example you cited. A multiplication of scalar to scalar will give you a scalar quanitity. But a multiplication of a Scalar with a Vector will result in a Vector quaniity. Do you know why? Here is a way to understand it. In mathematics when we multiply any thing by One, the same number returns. In this case a Scalar quantity can be written as
Scalar * 1 * Vector Magnitude * by its Direction.
So it will return the Direction multiplied by 1 itself hence retaining "Direction". In other words the result has a "magnitude as well as a direction" But isn't such a quantity we call a Vector.