Algebraic Vectors

By placing geometric vectors in the context of a rectangular coordinate system,
we can think of vectors in an algebraic way
Then we can employ all the operations of algebra to work abstractly with vectors
and use vectors in the solutions of problems

                    

We will denote a vector by putting an arrow over the symbol
For example, in the figure above, the boldface vector will be written as

     See Example 1, page 604, of the textbook


The magnitude, or norm, of a vector is given by

          

     See Example 2, page 605, of the textbook


                    Vector Addition
If   and  ,  then
                                                                                     


               Scalar Multiplication of a Vector
If    and    is a scalar, then
                                                                                     

     See Example 3, page 606, of the textbook



               Unit Vectors
A vector whose magnitude, or norm, is equal to 1 is called a unit vector

If    is a nonzero vector, then
                                                                 
is a unit vector with the same direction as


     See Example 4, page 607, of the textbook


                                             The and unit vectors.

          

           is a unit vector parallel to the x-axis           is a unit vector parallel to the y-axis

These unit vectors are important because any vector
                    
can be written as a linear combination of these two vectors

     

         

        

     See Examples 5, page 608, of the textbook




          Summary of Properties


For all vectors , , and all scalars and ,
the following properties hold

          Addition Properties

1.                                               Commutative Property

2.                  Associative Property

3.                                     Additive Identity

4.                    Additive Inverse


          Scalar Multiplication Properties

1.                                   Associative Property

2.                        Distributive Property

3.                       Distributive Property

4.                                                         Multiplicative Identity


     See Example 6, page 609, of the textbook

Static Equilibrium – see Example 7, pages 609 – 610, of the textbook


                    top
                    next  Polar Coordinates and Graphs