Vectors in the Plane

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= 3,4 . 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