The vector notation amounts simply to an abbreviation for writing the vector's three components. In two dimensions, a vector can be represented either by its two components or by its magnitude and direction. The two ways of describing a vector can be related by trigonometry. The two main operations on vectors are addition of a vector to a vector, and multiplication of a vector by a scalar. Vector addition means adding the components of two vectors to form the components of a new vector.
Vector subtraction is performed by negating the vector to be subtracted and then adding.
Multiplying a vector by a scalar means multiplying each of its components by the scalar to create a new vector. Division by a scalar is defined similarly. The Adobe Acrobat version of the book contains the homework problems.