There are times when we will have to multiply two vectors together to define some third quantity. The strange thing is, there are two ways we can multiply vectors - one where we get a scalar for an answer, and a second where we get a new vector for an answer. These are called dot (or scalar) products and cross (or vector) products, respectively.
Dot products have two ways of being calculated, and choose the method that makes it easier based on the information given in the problem. Cross products are a bit more involved and require a 3x3 determinant to figure out the x, y, and z components of the new vector.
Geometrically, dot products multiply the two parallel components of the two vectors. The formula, if we know the magnitudes of the two vectors, is A dot B = ABcos(theta). And cross products multiply the two perpendicular components of the two vectors, or A x B = ABsin(theta). In these definitions, theta is the angle between vectors A and B.