6.3 Vectors in the Plane

Inroduction

Many quantities can be represented by a single numbers. Others need both magnitude and direction. These numbers are called Vectors. Vectors can be represented by a directed line segment. A directed line segment has an initial and terminal point, both represented by a letter. 

Vectors Representation by Directed Line Segments

Example: let u be represented by by points a=(0,0) and b=(4,3). Let v be represented by points c=(1,4) and d=(5,7). Show that u=v.

Solution: use the distance formulas to find the magnitude. 

  

Both of the segments also have a slope of 3/4 so u=v

Component Form of a Vector

When a directed line segment has its initial point on the origin, it is known to be in standard position.

A vector in standard form can be uniquely represented by its terminal point, which is known as Component Form. 

The component form of initial point                            and terminal point                                                           

                                                                                  is 

Then the magnitude is found by using the distance formula.

Vector Operations

The two basic operations are scalar multiplication and vector addition. 

For scalar multiplication, the product of vector v and scalar |k| is a vector that is k times as long as v. 

or

For vector addition, the sum of u and v can be found by

Vector Operations

Problem: v=(-1,4) and w= (3,2). Find v+2w.

Solution: 

(-1,4)+2(3,2)=v+2w

(-1,4)+(6,4)=

(5,8)=

Properties of vectors include:

Many normal properties of addition and multiplication also apply to vectors

u+v=v+u

(u+v)=w=u+(w+v)

u+0=u

u+(-u)=0

c(du)=(cd)u

(c+d)u=cu+du

1(u)=u and 0(u)=0

||cv||= |c| ||v||

Unit Vector

the standard unit vectors are (1,0) and (0,1). i=(1,0) and j=(0,1), with i being the horizontal component and j being the vertical component.

Problem: write (4,7) in vector form.

Solution: 4i+7j 

the vector can be switched back and forth easily from vector form.

Vector form is 

Direction Angles

The unit vector shown below is rotated about the origin. If you were to go counterclockwise from the x-axis, you could find the Direction angle of the vector. the direction angle is the angle made by the vector and the x-axis.