Unit 6.1

Copyright © 2011 Pearson, Inc.

6.1Vectors in the

Plane

Copyright © 2011 Pearson, Inc. Slide 6.1 - 2

What you’ll learn about

Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors

… 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 two - dimensional vector v is an ordered pair of real

numbers, denoted in component form as a,b . The

numbers a and b are the components of the vector v.

The standard representation of the vector a,b is the

arrow from the origin to the point (a,b). The magnitude

of v is the length of the arrow and the direction of v is the

direction in which the arrow is pointing. The vector

0 = 0,0 , called the zero vector, has zero length and

no direction.


Initial Point, Terminal Point, Equivalent


Head Minus Tail (HMT) Rule

If an arrow has initial point x1, y

1 and terminal point

x2, y

2 , it represents the vector x2 x

1, y

2 y

1.


Magnitude

If v is represented by the arrow from x1, y

1 to x2, y

2 ,then

v x2 x

1 2 y

2 y

1 2.

If v a,b , then v a2 b2 .


Example Finding Magnitude of a Vector

Find the magnitude of v represented by PQ,

where P (3, 4) and Q (5,2).


Example Finding Magnitude of a Vector

v x2 x

1 2 y

2 y

1 2

5 3 2 2 ( 4) 2

2 10

Find the magnitude of v represented by PQ,

where P (3, 4) and Q (5,2).


Vector Addition and Scalar Multiplication

Let u u1,u

2 and v v

1,v

2 be vectors and let k be

a real number (scalar). The sum (or resultant) of the

vectors u and v is the vector

u v u1 v

1,u

2 v

2.

The product of the scalar k and the vector u is

ku k u1,u

2 ku

1,ku

2.


Example Performing Vector Operations

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


Example Performing Vector Operations

3u 3 2 , 3 1 = 6, 3

3u v = 6, 3 5,3 6 5, 33 11,0

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


Unit Vectors

A vector u with | u | 1 is a unit vector. If v is not

the zero vector 0,0 , then the vector u v

| v |

1

| v |v

is a unit vector in the direction of v.


Example Finding a Unit Vector

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


Example Finding a Unit Vector

v 2, 3 22 3 2 13, so

v

v

1

132, 3

2

13,

3

13

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


Standard Unit Vectors

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

unit vectors. Any vector v can be written as an expression

in terms of the standard unit vector:

v a,b

a,0 0,b

a 1,0 b 0,1

ai bj


Resolving the Vector

If v has direction angle , the components of v can

be computed using the formula

v = v cos , v sin .

From the formula above, it follows that the unit vector

in the direction of v is u v

v cos ,sin .


Example Finding the Components of a Vector

Find the components of the vector v with

direction angle 120o and magnitude 8.


Example Finding the Components of a Vector

v a,b 8cos120o,8sin120o

8 1

2

,83

2

4,4 3

So a 4 and b 4 3.

Find the components of the vector v with

direction angle 120o and magnitude 8.


Example Finding the Direction Angle of a Vector

Let u 2,3 and v 4, 1 .

Find the magnitude and direction angle of each vector.



Let be the direction angle of u, then

| u | 2 232 13

2 u cos

2 13cos

cos 2

13

Let u 2,3 and v 4, 1 .


cos 1 2

13

90¼ 180¼

123.69¼



Let be the direction angle of v, then

| v | 4 2 1 2

17

4 v cos

4 17 cos

cos 4

17

Let u 2,3 and v 4, 1 .


360¼ cos 1 4

17

180¼ 2700¼

194.04¼



Interpret

The direction angle for u

is 123.69¼.

The direction angle for v

is 194.04¼.

Let u 2,3 and v 4, 1 .



Velocity and Speed

The velocity of a moving object is a vector

because velocity has both magnitude and

direction. The magnitude of velocity is speed.


Quick Review

1. Find the values of x and y.

2. Solve for in degrees. sin-1 3

11


Quick Review

3. A naval ship leaves Port Northfolk and averages

43 knots (nautical mph) traveling for 3 hr on a bearing

of 35o and then 4 hr on a course of 120o.

What is the boat's bearing and distance from

Port Norfolk after 7 hr.

The point P is on the terminal side of the angle .

Find the measure of if 0o 360o.

4. P(5,7)

5. P(-5,7)


Quick Review Solutions

1. Find the values of x and y.

x 7.5, y 7.5 3

2. Solve for in degrees. sin-1 3

11

64.8¼


Quick Review Solutions

3. A naval ship leaves Port Northfolk and averages

43 knots (nautical mph) traveling for 3 hr on a bearing

of 35o and then 4 hr on a course of 120o.

What is the boat's bearing and distance from

Port Norfolk after 7 hr.

distance = 224.2; bearing = 84.9¼

The point P is on the terminal side of the angle .

Find the measure of if 0o 360o.

4. P(5,7) 54.5¼

5. P( 5,7) 125.5¼

Education

Unit 6.1