A scalar is simply a value or a number.
Vector notation varies, and often depends upon the context.
We will only concern ourselves with vectors that are well defined in some orthogonal or perpendicular set of coordinate axes. Thus, vectors will be defined by an ordered set of numbers. In a Cartesian system, those would be the usual x, y, and z. The dot product of a vector with itself results in the norm or the square of the vector's magnitude. The Cross Product is slightly more complicated, but also a powerful concept.