6.36.3Vectors in the PlaneVectors in the Plane

Quick Review

-1

1. Find the values of and .

32. Solve for in degrees. sin11

3. A naval ship leaves Port Northfolk and averages 43 knots (nauticalmph) traveling for 3 hr on a bearing of 35 and then 4 h

x y

r on a courseof 120 . What is the boat's bearing and distance from Port Norfolkafter 7 hr.

Quick Review Solutions

-1

1. Find the values of and .

32. Solv

7.5, 7.5 3

64.8e for in degrees. sin 11

3. A naval ship leaves Port Northfolk and averages 43

x y

x y

knots (nauticalmph) traveling for 3 hr on a bearing of 35 and then 4 hr on a courseof 120 . What is the boat's bearing and distance from Port N

distance=orfolk

a 224.2; fter 7 bearin ghr. =84.9

What you’ll learn about• How to represent vectors as directed line segments• How to perform basic Vector Operations• How to write vectors as linear combinations of Unit

Vectors• How to find the Direction Angles of vectors• How to use vectors to model and solve real-life

problems… and whyThese topics are important in many real-world

applications, such as calculating the effect of the wind on an airplane’s path.

Directed Line Segment

Two-Dimensional Vector

A is an ordered pair of real numbers, denoted in as , . The numbers and are

the of the vector . The of the vector ,

a b a b

a b

two - dimensional vectorcomponent form

components standard representation

v

vis the arrow from the origin to the point ( , ).

The of is the length of the arrow and the of is the direction in which the arrow is pointing. The vector

= 0,0 , called the

a b

magnitude direction

0 ze

vv

, has zero length and no direction.ro vector

Initial Point(R), Terminal Point(S), Equivalent(P)

Magnitude

1 1 2 2

2 2

2 1 2 1

2 2

If is represented by the arrow from , to , , then

.

If , , then .

x y x y

v x x y y

a b a b

v

v v

Example Finding Magnitude of a Vector

Find the magnitude of represented by , where (3, 4) and

(5, 2).PQ P

Q

v

Example Finding Magnitude of a Vector

Find the magnitude of represented by , where (3, 4) and

(5, 2).PQ P

Q

v

2 2

2 1 2 1

2 2

5 3 2 ( 4)

2 10

x x y y

v

Vector Addition and Scalar Multiplication

1 2 1 2

1 1 2 2

Let , and , be vectors and let be a real number

(scalar). The (or ) is the vector, .

The and the vector is

u u v v k

u v u v

k k u

sum resultant of the vectors and

product of the scalar

u v

u vu v

k uu

1 2 1 2, , .u ku ku

Example Performing Vector Operations

Let 2, 1 and 5,3 . Find 3 . u v u v

Example Performing Vector Operations

Let 2, 1 and 5,3 . Find 3 . u v u v

3 3 2 , 3 1 = 6, 3

3 = 6, 3 5,3 6 5, 3 3 11,0

u

u v

Unit Vectors

A vector with || || 1 is a . If is not the zero vector10,0 , then the vector is a

|| || || ||.

unit vector

unit vector in the direction

of

u u vvu vv v

v

Example Finding a Unit Vector

Find a unit vector in the direction of 2, 3 . v

Example Finding a Unit Vector

Find a unit vector in the direction of 2, 3 . v

222, 3 2 3 13, so

1 2 32, 3 , 13 13 13

| |

| |

v

v

v

Standard Unit Vectors

The two vectors 1,0 and 0,1 are the standard

unit vectors. Any vector can be written as an expressionin terms of the standard unit vector:

,

,0 0,

1,0 0,1

a b

a b

a b

a b

i j

v

v

i j

Resolving the Vector If has direction angle , the components of can be computed

using the formula = | | cos , | | sin .

From the formula above, it follows that the unit vector in the

direction of is cos ,sin .| |

v v

v v v

vv uv

Example Finding the Components of a

Vector

Find the components of the vector with direction angle 120 andmagnitude 8.

v

Example Finding the Components of a

Vector

Find the components of the vector with direction angle 120 andmagnitude 8.

v

, 8cos120 ,8sin120

1 3 8 ,82 2

4,4 3

So 4 and 4 3.

a b

a b

v

Example Finding the Direction Angle of a Vector

Find the magnitude and direction angle of 2,3 .u

Example Finding the Direction Angle of a Vector

Find the magnitude and direction angle of 2,3 .u

2 2|| || 2 3 13Let be the direction angle of , then

2,3 13 cos , 13sin

2 13 cos56.3

uu

u

Velocity and SpeedThe velocity of a moving object is a

vector because velocity has both magnitude

and direction. The magnitude of velocity is

speed.