The useful vector operations, such as addition and scalar multiplication, are rotationally invariant, i.e. come out the same regardless of the orientation of the coordinate system. 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.