Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Section 6.6 Vectors.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Section 6.6

Vectors


Objectives:

• Use magnitude and direction to show vectors are equal.• Visualize scalar multiplication, vector addition, and

vector subtraction as geometric vectors.• Represent vectors in the rectangular coordinate system.• Perform operations with vectors in terms of i and j.• Find the unit vector in the direction of v.• Write a vector in terms of its magnitude and direction.• Solve applied problems involving vectors.


Vectors

Quantities that involve both a magnitude and a direction are called vector quantities, or vectors for short.

Quantities that involve magnitude, but no direction, are called scalar quantities, or scalars for short.


Directed Line Segments and Geometric Vectors

A line segment to which a direction has been assigned is called a directed line segment. We call P the initial point and Q the terminal point. We denote this directed line segment by .PQ

��

The magnitude of the directed line

segment is its length. We

denote this by Length can’t be

negative so magnitude is not negative.

Geometrically, a vector is a directed

line segment.

PQ��

.PQ��


Representing Vectors in Print and on Paper

In Print written in bold.

On paper write with arrow over letter.


Equal Vectors

In general, vectors v and w are equal if they have the same magnitude and the same direction. We write this v = w.


Example: Showing that Two Vectors are Equal

Show that u = v.

Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude.

222 1 2 1( )u x x y y

2 2(6 2) [ 2 ( 5)] 2 24 3

16 9 25 5



Show that u = v.

Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude.

222 1 2 1( )v x x y y

22(6 2) 5 2

2 24 3

16 9 25 5



Show that u = v.

One way to show that u and v have the same direction is to find the slopes of the lines on which they lie.

2 1

2 1

y ym

x x

6 22 ( 5)

43

2 1

2 1

y ym

x x

6 25 2

43

slope of u

slope of v



Show that u = v.

5u

5v

slope of u slope of v43

43

The vectors have the same magnitudeand direction. Thus, u = v.


Scalar Multiplication

The multiplication of a real number k and a vector v is called scalar multiplication. We write this product kv.

Multiplying a vectorby any positive realnumber (except 1)changes the magnitudeof the vector but not its direction.

Multiplying a vector by any negativenumber reversesthe direction of the vector.


Scalar Multiplication (continued)


The Sum of Two Vectors

The sum of u and v, denoted u + v is called the resultant vector. A geometric method for adding two vectors is shown in the figure. Here is how we find this vector:

•Position u and v, so thatthe terminal point of ucoincides with the initialpoint of v.

•The resultant vector, u + v, extends from the initial point of u to theterminal point of v.


The Difference of Two Vectors

The difference of two vectors, v – u, is defined as v – u = v + (–u), where –u is the scalar multiplication of u and –1, –1u. The difference v – u is shown geometrically in the figure.


The i and j Unit Vectors


Representing Vectors in Rectangular Coordinates


Example: Representing a Vector in Rectangular Coordinates and Finding Its Magnitude

Sketch the vector v = 3i – 3j and find its magnitude.v ai vj

3 3v i j 3, 3a b

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

initial point(0, 0)

terminal point

(3, –3)

v = 3i – 3j

222 1 2 1( )v x x y y

2 23 ( 3)

9 9 18 3 2


Representing Vectors in Rectangular Coordinates (continued)

A vector with its initial point at the origin is called a position vector.

If a vector’s initial point is not at the origin, it can be shown to be equal to a position vector.


Example: Representing a Vector in Rectangular Coordinates

Let v be the vector from initial point P1 = (–1, 3) to terminal point P2 = (2, 7). Write v in terms of i and j.

2 1 2 1( ) ( )v x x i y y j

[2 ( 1)] (7 3)i j

3 4i j -4 -3 -2 -1 1 2 3 4

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

v = 3i + 4j

1 ( 1,3)P

2 (2,7)P

(3,4)


Adding and Subtracting Vectors in Terms of i and j


Example: Adding and Subtracting Vectors

If v = 7i + 3j and w = 4i – 5j, find the following vectors:

a. v + w

b. v – w

1 2 1 2( ) ( )v w a a i b b j

(7 4) [3 ( 5)]i j

11 2i j

1 2 1 2( ) ( )v w a a i b b j

(7 4) [3 ( 5)]i j

3 8i j


Scalar Multiplication with a Vector in Terms of i and j


Example: Scalar Multiplication

If v = 7i + 10j, find each of the following vectors:

a. 8v

b. –5v

( ) ( )kv ka i kb j 8 (8 7) (8 10)v i j

56 80i j

( ) ( )kv ka i kb j

5 ( 5 7) ( 5 10)v i j 35 50i j


The Zero Vector


Properties of Vector Addition


Properties of Scalar Multiplication


Unit Vectors

A unit vector is defined to be a vector whose magnitude is one.


Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v


Example: Finding a Unit Vector

Find the unit vector in the same direction as v = 4i – 3j. Then verify that the vector has magnitude 1.

2 2v a b 2 24 ( 3) 16 9 25 5

4 35

v i jv

2 24 35 5

4 35 5

i j

16 925 25

251

25


Writing a Vector in Terms of Its Magnitude and Direction


x

y

Example: Writing a Vector Whose Magnitude and Direction are Given

The jet stream is blowing at 60 miles per hour in the direction N45°E. Express its velocity as a vector v in terms of i and j.

v

45 , 60v cos sinv v i v j

60cos45 60sin 45v i j 2 2

60 602 2

i j

30 2 30 2i j

The jet stream can be expressed in terms of i and j as

30 2 30 2i j


Example: Application

Two forces, F1 and F2, of magnitude 30 and 60 pounds, respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E


Example: Application (continued)

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E

1 1 1cos sinF F i F j 30cos80 30sin80i j 5.21 29.54i j

2 2 2cos sinF F i F j 60cos30 60sin30i j 51.96 30i j


Example: Application (continued)

1 5.21 29.54F i j

2 51.96 30F i j

1 2F F F (5.21 29.54 ) (51.96 30 )i j i j (5.21 51.96) (29.54 30)i j 57.17 59.54i j

2 2F a b 2 257.17 59.54 82.54

cosaF

57.1782.54

1 57.17

cos82.54

46.2


Example: Application

Two forces, F1 and F2, of magnitude 30 and 60 pounds, respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.

x

y Resultantforce, F

F2

60 pounds

F1

30 pounds

10N E

60N E

The two given forces are equivalent to a single force ofapproximately 82.54 pounds with a direction angle of approximately 46.2°.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Section 6.6 Vectors.

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