Part 1: Vectors in the Plane

Now that we have studied calculus in a 2 dimensional setting, our next step is to extend calculus to 3 or more dimensions. However, a slope allows only a rise and a run, and as a result, slope is inherently a two dimensional concept. Thus, we begin with a concept that is the same across any number of dimensions -- the concept of a vector.

Given two points P₁( x₁,y₁) and P₂(x₂,y₂) in the xy-plane, we define the vector between them to be

v = P1P2  =  áx2-x1,y2-y1 ñ

(1)

The points P₁ and P₂ are called, respectively, the initial point and the terminal point of the vector v, and v is often represented by an arrow beginning at a point P₁ and ending at P₂.

In general, a scalar is a number such as 1, p, or 9.28, and a 2-dimensional vector is a grouping of two scalars a and b in the form v = áa,b ñ.  The scalars a and b are known as the components of v = á a,b ñ .       

EXAMPLE 1    Find the vector u with initial point P₁( 2,4) and final point P₂( 5,6) .

Solution: To find u, we apply (1) in the form

u =   P1P2   = á 5-2,6-4 ñ = á 3,2 ñ

       

EXAMPLE 2    Find the vector w with initial point P₁( 9,4) and final point P₂( 12,6) .

Solution: To do so, we apply (1) in the form

w =   P1P2   á 12-9,6-4 ñ = á 3,2 ñ

Notice that the vectors u and w in examples 1 and 2 are the same, as is further illustrated in the figure below:

That is, the translation of a vector results in the same vector, just as the translation of a line results in a line with the same slope as the original. 

A 2-dimensional vector v = áa,bñ can also be defined in terms of a magnitude ||v|| and a direction angle a using trigonometric relationships:

a = || v|| cos( a) b = || v|| sin( a)

We say that v = á || v|| cos(a), || v|| sin( a) ñ is the polar form of v.  The Pythagorean theorem says that

||v|| =   a2 + b2

and the quantity cos(a) = a / ||v|| is called a direction cosine of v.          

EXAMPLE 3    What is the magnitude and direction angle of

v = á 3,2 ñ

Solution: The magnitude is

||v|| =   32 + 22   =   13

The direction angle follows from the direction cosine:

cos(a) =    3 13 ,   so a = cos-1 æ ç è 3 13   ö ÷ ø  = 0.588 radians

In degree measure, the direction angle is

a = 0.588 radians  · æ è 180° p radians ö ø  = 33.6899°

to 4 decimal places.

It follows that vectors with the same direction angle are parallel and that vectors with the same magnitude and direction are