V = { { (x,y) with x and y real numbers } and addition and multiplication are as follows

() + () = () , k()=()}
where k is any real number ( scalar ).

This is a vector space because we can show that the ten axioms for a vector space are satisfied.We could refer to this as the vector space of ordered pairs. Your text refers to it as the space of vectors in a plane because of the picture( graph ) we will use for this space. We can embellish the space by defining more operations on it. We'll do this later. One of the operations will "induce" a norm on the space and we define the norm of a vector (x,y) as : 

The following idea is crucial in your work with these vectors. The graphic representation for a vector is an infinite set of directed line segments that have the same direction ( on parallel lines or on the same line ), magnitude ( the norms are the same; that is , have the same length), and sense ( arrow tip pointing in the same direction ). If we must draw an infinite set of arrow tipped lines each time we graph a vector, the picture will be less useful. Instead, we draw one representative for the entire set and call it a position vector.  It eminates from the origin. Any other arrow tipped line segment with the same direction, magnitude, and sense represents the same vector. We often speak of the representative as the vector.

This idea of many arrow tipped lines for the same ordered pair is nicely illustrated in our text as follows:

See 11.1 Example 1

See 11.1 Example1 Length

See 11.1 Example 1 Direction

Notice that when the vector representative eminates from the origin, the norm of the vector is the length of the line segment between (0,0) and (x,y). This explains my reference to length when "talking" about norm in the previous paragraph.

The idea that is crucial is that a vector may be drawn between many different points in the plane, but one orderd pair will be "attached" , so to speak, to all such line segments and is thus the algebraic representation for the vector.

See 11.1 Example 2

Following Example 2 your authors explain Vector Operations  It is important for you to observe two graphic representations for vector addition that I will refer to as : "tip to tail" and the "diagonal of a parallelogram" methods. Notice in the "diagonal" representation" the addition vector ( "resultant", your author calls it ) and the vectors being added all eminate from the same point. We'll examine a reason for using the word "resultant" for the vector sum at a later time.

The authors also define the subtraction of two vectors and we find the vector for that operation across the tips of the two vectors being subtracted if they eximate from the same point. It is relatively easy to "see" that the vector (u - v) placed across the tips of u and v in the correct direction produces u by the "tip to tail" method. We expect (u - v) + v to result in u by the axioms of our vector space.

Lastly, the authors "show" scalar multiplication as "stretching" or "shrinking" ( this depends on whether the multiplier is greater or less than the number 1 ) and that multiplying by a negative number also changes the sense (given by the arrow tip).

Having illustrated the graphic interpretations of the vector operations the authors hasten to remind us that we are in a vector space because the ten defining properties for a space are satisfied and they embellish with a few proofs which we consider optional at this time. However, the development of the process called "normalizing a vector" is not optional and will serve us repeatedly throughtout our study of these vectors. Now carefully, and if necessary, repeatedly read and study the following:

See 11.1 Vector Operations (following Example 2 )

See 11.1 Example 3

The idea we referred to as "normalizing a vector" will have many uses and we begin with the authors' next few examples that involve unit vectors:

See 11.1 Example 4

See 11.1 Standard Unit Vectors

See 11.1 Example 5

See 11.1 Circle ( between Ex.5 and Ex.6

see 11.1 Example 6

Example 6 illustrates and emphasizes an idea we are are aware of: If a vector is represented by an arrow- tipped line then a circle with radius 1 can be used in describing unit vectors; and, if the norm of the vector is not 1, a non unit circle can be used. This idea is crucial to the first application problems we do using vectors. In these next problems you'll see how information often comes to us in terms of Trigonometry, which we recognize as involving a circle, and that forces "act" in ways that can be described using the mathematics of vectors. The Physicist through experiments establishes for us the "Laws" of forces and the mathematics of vectors is appropriate for describing and for computing results for problems in this "field".

Using Vector Mathematics:

See 11.1 Example 7

See 11.1 Example 8

Look at the following solved problems in the Exercises at the end of 11.1 : 1,9,17,21,27,29,33,53,77 Then do the following unsolved problems as homework: 4,10,18,22,26,30,34,54,78