1. Vectors in the Plane
Peter Paliwoda

2. Introduction to Vectors
Quantities such as force and velocity involve both magnitude and direction
Such quantities cannot be characterized by single real numbers
To represent such a quantity, you can use a directed line segment shown in Fig. 6.15

3. Introduction to Vectors
The directed line segment PQ has initial point P and terminal point Q. Its magnitude or (length) is denoted by found by Distance Formula

4. Introduction to Vectors
Two directed line segments that have same magnitude and direction are equivalent
For example, segments shown below in Figure 6.16 are all equivalent
Vectors are denoted by lowercase, boldface letters such as u, v, and w. Ex. v= PQ

5. Introduction to Vectors
Let u represent line segment from P=(0,0) to Q=(3,2)
Let v represent line segment from R=(1,2) to S=(4,4)
Show that u=v
Moreover, both lines have the same direction because they are both directed toward upper right on lines having slope of 2/3. Therefore PQ and RS have the same magnitude and direction, thus u=v

6. Component Form of a Vector
A vector whose initial point is the origin (0,0) can be uniquely represented by the coordinates of its terminal point (v1, v2)
This is the component form of a vector v, written as v= v1, v2

7. Finding the Component Form of a Vector
Initial point (4,-7) and terminal point (-1,5)
v1=-1-4=-5; v2=5-(-7)=12, so v= -5,12
So v= -5,12 and the magnitude of v is

8. Vector Operations
Two basic operations are scalar multiplication and vector addition

9. Vector Operations
Let v= -2,5 and w= 3, Find 2v, w-v, v+2w

11. Unit Vectors
In many applications its useful to find a unit vector that has the same direction as a given nonzero vector v.
Example, find the unit vector of v= -2,5

12. Example Problem