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Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4
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��������������
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5
Representing Vectors in Print and on Paper
In Print written in bold.
On paper write with arrow over letter.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8
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( )v x x y y
22(6 2) 5 2
2 24 3
16 9 25 5
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9
Example: Showing that Two Vectors are Equal
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10
Example: Showing that Two Vectors are Equal
Show that u = v.
5u
5v
slope of u slope of v43
43
The vectors have the same magnitudeand direction. Thus, u = v.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16
Representing Vectors in Rectangular Coordinates
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19
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)
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20
Adding and Subtracting Vectors in Terms of i and j
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22
Scalar Multiplication with a Vector in Terms of i and j
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27
Unit Vectors
A unit vector is defined to be a vector whose magnitude is one.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28
Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30
Writing a Vector in Terms of Its Magnitude and Direction
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34
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
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35
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°.