VECTORS IN THE PLANE

OBJECTIVES:• Represent vectors as directed line segments• Write the component forms of vectors• Perform basic vector operations and represent them graphically• Write vectors as linear combinations or unit vectors• Find the direction angles of vectors• Use vectors to model and solve real-life problems

Introduction Quantities such as force and velocity involve both

magnitude and direction and cannot be completely characterized by a single real number. To represent such a quantity you can use a directed line segment: a line segment with an initial point and terminal point. Its magnitude (or length) is denoted by and can be found using the Distance Formula.

Two directed line segments that have the same magnitude and direction are equivalent.

Directed line segments are vectors in the plane

PQ

Showing that two vectors are equivalent

EX 1: Show that in the figure are equivalent

and u v

0,0P

3,1Q 2,2R

5,3S

u

v

Component Form of a Vector Standard position: a vector whose initial point is

the origin.

Component Form of a Vector: The component form of the vector with initial point

and terminal point is given by

The magnitude (or length) of v is given by

1 2,P p p 1 2,Q q q

1 1 2 2 1 2, ,PQ q p q p v v v

2 2 2 21 1 2 2 1 2v q p q p v v

Finding the Component Form of a Vector

EX 2: Find the component form and magnitude of the vector v that has initial point and terminal point

4, 7 1,5

Vector operations The two basic vector operations are scalar

multiplication and vector addition. In operations with vectors, numbers are usually referred to as scalars.

Definitions of Vector Addition and Scalar Multiplication Let be vectors and let be

a scalar (a real number). Then the sum of is the vector

And the scalar multiple of times is the vector

1 2 1 2, and ,u u u v v v k

k

and u v

u1 1 2 2,u v u v u v

1 2 1 2, ,ku k u u ku ku

Vector Operations EX 3: Let . Find each of

the following vectors. A)

B)

2,5 and 3,4v w

2v

v w

Vector Operations Continue EX 3

C)

D)

w v

2v w

Unit Vectors In many applications of vectors, it is useful to find

a unit vector that has the same direction as a given nonzero vector v.

To do this, you can divide v by its magnitude to obtain

Note that u is a scalar multiple of v. The vector u has a magnitude of 1 and the same direction as v. The vector u is called a unit vector in the direction of v.

1unit vector vu vv v

Finding a Unit Vector EX 4: Find a unit vector in the direction of

and verify that the result has a magnitude of 1 2,5v

Finding a Vector EX 5: Find the vector w with magnitude

and the same direction as 5w

2,3v

Linear Combination of Vectors The unit vectors are called

standard unit vectors and are denoted by

These vectors can be used to represent any vector as follows:

The scalars are called the horizontal and vertical components of v, respectively.

The vector sum is called a linear combination of the vectors

1,0 and 0,1

1,0 and 0,1i j

1 2,v v v 1 2

1 2

1 2

,

1,0 0,1

v v v

v v

v i v j

1 2 and v v

and i j1 2v i v j

Writing a Linear Combination of Unit Vectors

EX 6: Let be the vector with initial point and terminal point . Write as a linear combination of the standard unit vectors

EX 7:

uu

and i j

2, 5 1,3

Let 3 8 and 2 . Find 2 3u i j v i j u v

Direction Angles If is a unit vector such that is the angle

(measured counterclockwise) from the positive x-axis to , then the terminal point of lies on the unit circle and you have

The angle is the direction angle of the vector .

u

u u

,

cos ,sin

cos sin

u x y

i j

u

Direction Angles Suppose that is a unit vector with direction angle

If is any vector that makes an angle with the positive x-axis, then it has the same direction as and you can write

Because , it follows that the direction angle for is determined from

u v ai bj

u cos ,sin cos sinv v v i v j cos sinv ai bj v i v j

vsintancos

sincos

v bav

Finding Direction Angles of Vectors

EX 8: Find the direction angle of each vector

A)

B)

3 3u i j

3 4v i j

Using Vectors to find Speed and Direction

An airplane is traveling at a speed of 500 miles per hour with a bearing of at a fixed altitude with a negligible wind velocity. When the airplane reaches a certain point, it encounters a wind with a velocity of 70 miles per hour in the direction . What are the resultant speed and direction of the airplane?

330

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