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Precalculus

8.8 Vectors

Precalculus8.8 Vectors
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  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section you will:
  • View vectors geometrically.
  • Find magnitude and direction.
  • Perform vector addition and scalar multiplication.
  • Find the component form of a vector.
  • Find the unit vector in the direction of vv.
  • Perform operations with vectors in terms of ii and jj.
  • Find the dot product of two vectors.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in Figure 1. What are the ground speed and actual bearing of the plane?

Image of a plan flying SE at 140 degrees and the north wind blowing
Figure 1

Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

A Geometric View of Vectors

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:

  • Lower case, boldfaced type, with or without an arrow on top such as v,u,w, v , u , w . v,u,w, v , u , w .
  • Given initial point P Pand terminal point Q, Q,a vector can be represented as PQ . PQ .The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
  • Given an initial point of ( 0,0 ) ( 0,0 )and terminal point ( a,b ), ( a,b ),a vector may be represented as a,b . a,b .

This last symbol a,b a,b has special significance. It is called the standard position. The position vector has an initial point ( 0,0 ) ( 0,0 )and a terminal point a,b . a,b .To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector CD CD is C( x 1 , y 1 ) C( x 1 , y 1 )and the terminal point is D( x 2 , y 2 ), D( x 2 , y 2 ),then the position vector is found by calculating

AB = x 2 x 1 , y 2 y 1 = a,b AB = x 2 x 1 , y 2 y 1 = a,b

In Figure 2, we see the original vector CD CD and the position vector AB . AB .

Plot of the original vector CD in blue and the position vector AB in orange extending from the origin.
Figure 2

Properties of Vectors

A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at ( 0,0 ) ( 0,0 )and is identified by its terminal point a,b . a,b .

Example 1

Find the Position Vector

Consider the vector whose initial point is P( 2,3 ) P( 2,3 )and terminal point is Q( 6,4 ). Q( 6,4 ).Find the position vector.

Example 2

Drawing a Vector with the Given Criteria and Its Equivalent Position Vector

Find the position vector given that vector v vhas an initial point at ( 3,2 ) ( 3,2 )and a terminal point at ( 4,5 ), ( 4,5 ),then graph both vectors in the same plane.

Try It #1

Draw a vector v vthat connects from the origin to the point (3,5). (3,5).

Finding Magnitude and Direction

To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

Magnitude and Direction of a Vector

Given a position vector v v = a,b , = a,b ,the magnitude is found by | v |= a 2 + b 2 . | v |= a 2 + b 2 .The direction is equal to the angle formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction is found by tanθ=( b a )θ= tan 1 ( b a ), tanθ=( b a )θ= tan 1 ( b a ),as illustrated in Figure 5.

Standard plot of a position vector (a,b) with magnitude |v| extending into Q1 at theta degrees.
Figure 5

Two vectors v and u are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.

Example 3

Finding the Magnitude and Direction of a Vector

Find the magnitude and direction of the vector with initial point P( 8,1 ) P( 8,1 )and terminal point Q( 2,5 ). Q( 2,5 ).Draw the vector.

Example 4

Showing That Two Vectors Are Equal

Show that vector v with initial point at ( 5,−3 ) ( 5,−3 )and terminal point at ( −1,2 ) ( −1,2 )is equal to vector u with initial point at ( −1,−3 ) ( −1,−3 )and terminal point at ( −7,2 ). ( −7,2 ).Draw the position vector on the same grid as v and u. Next, find the magnitude and direction of each vector.

Performing Vector Addition and Scalar Multiplication

Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector uu = x,y = x,y as an arrow or directed line segment from the origin to the point (x,y), (x,y),vectors can be situated anywhere in the plane. The sum of two vectors u and v, or vector addition, produces a third vector u+ v, the resultant vector.

To find u + v, we first draw the vector u, and from the terminal end of u, we drawn the vector v. In other words, we have the initial point of v meet the terminal end of u. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in Figure 8.

Diagrams of vector addition and subtraction.
Figure 8

Vector subtraction is similar to vector addition. To find uv, view it as u + (−v). Adding −v is reversing direction of v and adding it to the end of u. The new vector begins at the start of u and stops at the end point of −v. See Figure 9 for a visual that compares vector addition and vector subtraction using parallelograms.

Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.
Figure 9

Example 5

Adding and Subtracting Vectors

Given u u = 3,2 = 3,2 and v v = −1,4 , = −1,4 ,find two new vectors u + v, and uv.

Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

Scalar Multiplication

Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply v v = a,b = a,b by k k, we have

kv= ka,kb kv= ka,kb

Only the magnitude changes, unless k kis negative, and then the vector reverses direction.

Example 6

Performing Scalar Multiplication

Given vector v v = 3,1 , = 3,1 ,find 3v, 1 2 1 2 v, v, and −v.

Analysis

Notice that the vector 3v is three times the length of v, 1 2 1 2 v vis half the length of v, and –v is the same length of v, but in the opposite direction.

Try It #2

Find the scalar multiple 3 u u given uu = 5,4 . = 5,4 .

Example 7

Using Vector Addition and Scalar Multiplication to Find a New Vector

Given u u = 3,2 = 3,2 and v v = 1,4 , = 1,4 ,find a new vector w = 3u + 2v.

Finding Component Form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the x xdirection, and the vertical component is the y ydirection. For example, we can see in the graph in Figure 12 that the position vector 2,3 2,3 comes from adding the vectors v1 and v2. We have v1 with initial point ( 0,0 ) ( 0,0 )and terminal point ( 2,0 ). ( 2,0 ).

v 1 = 20,00 = 2,0 v 1 = 20,00 = 2,0

We also have v2 with initial point ( 0,0 ) ( 0,0 )and terminal point ( 0,3 ). ( 0,3 ).

v 2 = 00,30 = 0,3 v 2 = 00,30 = 0,3

Therefore, the position vector is

v= 2+0,3+0 = 2,3 v= 2+0,3+0 = 2,3

Using the Pythagorean Theorem, the magnitude of v1 is 2, and the magnitude of v2 is 3. To find the magnitude of v, use the formula with the position vector.

|v|= | v 1 | 2 +| v 2 | 2 = 2 2 + 3 2 = 13 |v|= | v 1 | 2 +| v 2 | 2 = 2 2 + 3 2 = 13

The magnitude of v is 13 . 13 .To find the direction, we use the tangent function tanθ= y x . tanθ= y x .

tanθ= v 2 v 1 tanθ= 3 2 θ= tan 1 ( 3 2 )=56.3° tanθ= v 2 v 1 tanθ= 3 2 θ= tan 1 ( 3 2 )=56.3°
Diagram of a vector in root position with its horizontal and vertical components.
Figure 12

Thus, the magnitude of v vis 13 13 and the direction is 56.3 56.3 off the horizontal.

Example 8

Finding the Components of the Vector

Find the components of the vector v vwith initial point ( 3,2 ) ( 3,2 )and terminal point ( 7,4 ). ( 7,4 ).

Finding the Unit Vector in the Direction of v

In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.

Unit vectors are defined in terms of components. The horizontal unit vector is written as ii = 1,0 = 1,0 and is directed along the positive horizontal axis. The vertical unit vector is written as j j = 0,1 = 0,1 and is directed along the positive vertical axis. See Figure 14.

Plot showing the unit vectors i=91,0) and j=(0,1)
Figure 14

The Unit Vectors

If v vis a nonzero vector, then v | v | v | v | is a unit vector in the direction of v. v.Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.

Example 9

Finding the Unit Vector in the Direction of v

Find a unit vector in the same direction as v v = −5,12 〉.= −5,12 〉.

Performing Operations with Vectors in Terms of i and j

So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of i and j.

Vectors in the Rectangular Plane

Given a vector v vwith initial point P=( x 1 , y 1 ) P=( x 1 , y 1 )and terminal point Q=( x 2 , y 2 ), Q=( x 2 , y 2 ), v is written as

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

The position vector from ( 0,0 ) ( 0,0 )to ( a,b ), ( a,b ),where ( x 2 x 1 )=a ( x 2 x 1 )=aand ( y 2 y 1 )=b, ( y 2 y 1 )=b,is written as v = ai + bj. This vector sum is called a linear combination of the vectors i and j.

The magnitude of v = ai + bj is given as | v |= a 2 + b 2 . | v |= a 2 + b 2 . See Figure 16.

Plot showing vectors in rectangular coordinates in terms of i and j. The position vector v (in orange) extends from the origin to some point (a,b) in Q1. The horizontal (ai) and vertical (bj) components are shown.
Figure 16

Example 10

Writing a Vector in Terms of i and j

Given a vector v vwith initial point P=( 2,−6 ) P=( 2,−6 )and terminal point Q=( −6,6 ), Q=( −6,6 ),write the vector in terms of i iand j. j.

Example 11

Writing a Vector in Terms of i and j Using Initial and Terminal Points

Given initial point P 1 =( 1,3 ) P 1 =( 1,3 )and terminal point P 2 =( 2,7 ), P 2 =( 2,7 ),write the vector v vin terms of i iand j. j.

Try It #3

Write the vector u uwith initial point P=( 1,6 ) P=( 1,6 )and terminal point Q=( 7,5 ) Q=( 7,5 )in terms of i iand j. j.

Performing Operations on Vectors in Terms of i and j

When vectors are written in terms of i iand j, j,we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

Adding and Subtracting Vectors in Rectangular Coordinates

Given v = ai + bj and u = ci + dj, then

v+u=( a+c )i+( b+d )j vu=( ac )i+( bd )j v+u=( a+c )i+( b+d )j vu=( ac )i+( bd )j

Example 12

Finding the Sum of the Vectors

Find the sum of v 1 =2i3j v 1 =2i3jand v 2 =4i+5j. v 2 =4i+5j.

Calculating the Component Form of a Vector: Direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using iandj. iandj.For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.

Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with | v | | v |replacing r. r.

Vector Components in Terms of Magnitude and Direction

Given a position vector v= x,y v= x,y and a direction angle θ, θ,

cosθ= x |v| and sinθ= y |v| x=|v|cosθ y=|v|sinθ cosθ= x |v| and sinθ= y |v| x=|v|cosθ y=|v|sinθ

Thus, v=xi+yj=| v |cosθi+| v |sinθj, v=xi+yj=| v |cosθi+| v |sinθj,and magnitude is expressed as | v |= x 2 + y 2 . | v |= x 2 + y 2 .

Example 13

Writing a Vector in Terms of Magnitude and Direction

Write a vector with length 7 at an angle of 135° to the positive x-axis in terms of magnitude and direction.

Try It #4

A vector travels from the origin to the point ( 3,5 ). ( 3,5 ).Write the vector in terms of magnitude and direction.

Finding the Dot Product of Two Vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.

The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.

Dot Product

The dot product of two vectors v= a,b v= a,b and u= c,d u= c,d is the sum of the product of the horizontal components and the product of the vertical components.

vu=ac+bd vu=ac+bd

To find the angle between the two vectors, use the formula below.

cosθ= v | v | u | u | cosθ= v | v | u | u |

Example 14

Finding the Dot Product of Two Vectors

Find the dot product of v= 5,12 v= 5,12 and u= −3,4 . u= −3,4 .

Example 15

Finding the Dot Product of Two Vectors and the Angle between Them

Find the dot product of v1 = 5i + 2j and v2 = 3i + 7j. Then, find the angle between the two vectors.

Example 16

Finding the Angle between Two Vectors

Find the angle between u= 3,4 u= 3,4 and v= 5,12 . v= 5,12 .

Example 17

Finding Ground Speed and Bearing Using Vectors

We now have the tools to solve the problem we introduced in the opening of the section.

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? See Figure 19.

Image of a plan flying SE at 140 degrees and the north wind blowing.
Figure 19

Media

Access these online resources for additional instruction and practice with vectors.

8.8 Section Exercises

Verbal

1.

What are the characteristics of the letters that are commonly used to represent vectors?

2.

How is a vector more specific than a line segment?

3.

What are i iand j, j,and what do they represent?

4.

What is component form?

5.

When a unit vector is expressed as a,b , a,b ,which letter is the coefficient of the i i and which the j? j?

Algebraic

6.

Given a vector with initial point ( 5,2 ) ( 5,2 )and terminal point ( 1,3 ), ( 1,3 ),find an equivalent vector whose initial point is ( 0,0 ). ( 0,0 ).Write the vector in component form a,b . a,b .

7.

Given a vector with initial point ( 4,2 ) ( 4,2 )and terminal point ( 3,3 ), ( 3,3 ),find an equivalent vector whose initial point is ( 0,0 ). ( 0,0 ).Write the vector in component form a,b . a,b .

8.

Given a vector with initial point ( 7,1 ) ( 7,1 )and terminal point ( 1,7 ), ( 1,7 ),find an equivalent vector whose initial point is ( 0,0 ). ( 0,0 ).Write the vector in component form a,b . a,b .

For the following exercises, determine whether the two vectors u uand v vare equal, where u uhas an initial point P 1 P 1 and a terminal point P 2 P 2 and v v has an initial point P 3 P 3 and a terminal point P 4 P 4 .

9.

P 1 =( 5,1 ), P 2 =( 3,2 ), P 3 =( 1,3 ), P 1 =( 5,1 ), P 2 =( 3,2 ), P 3 =( 1,3 ),and P 4 =( 9,4 ) P 4 =( 9,4 )

10.

P 1 =( 2,3 ), P 2 =( 5,1 ), P 3 =( 6,1 ), P 1 =( 2,3 ), P 2 =( 5,1 ), P 3 =( 6,1 ),and P 4 =( 9,3 ) P 4 =( 9,3 )

11.

P 1 =( 1,1 ), P 2 =( 4,5 ), P 3 =( 10,6 ), P 1 =( 1,1 ), P 2 =( 4,5 ), P 3 =( 10,6 ),and P 4 =( 13,12 ) P 4 =( 13,12 )

12.

P 1 =( 3,7 ), P 2 =( 2,1 ), P 3 =( 1,2 ), P 1 =( 3,7 ), P 2 =( 2,1 ), P 3 =( 1,2 ),and P 4 =( 1,4 ) P 4 =( 1,4 )

13.

P 1 =( 8,3 ), P 2 =( 6,5 ), P 3 =( 11,8 ), P 1 =( 8,3 ), P 2 =( 6,5 ), P 3 =( 11,8 ),and P 4 =( 9,10 ) P 4 =( 9,10 )

14.

Given initial point P 1 =( 3,1 ) P 1 =( 3,1 )and terminal point P 2 =( 5,2 ), P 2 =( 5,2 ),write the vector v vin terms of i iand j. j.

15.

Given initial point P 1 =( 6,0 ) P 1 =( 6,0 )and terminal point P 2 =( 1,3 ), P 2 =( 1,3 ),write the vector v vin terms of i iand j. j.

For the following exercises, use the vectors u = i + 5j, v = −2i− 3j,  and w = 4ij.

16.

Find u + (vw)

17.

Find 4v + 2u

For the following exercises, use the given vectors to compute u + v, uv, and 2u − 3v.

18.

u= 2,3 ,v= 1,5 u= 2,3 ,v= 1,5

19.

u= 3,4 ,v= 2,1 u= 3,4 ,v= 2,1

20.

Let v = −4i + 3j. Find a vector that is half the length and points in the same direction as v. v.

21.

Let v = 5i + 2j. Find a vector that is twice the length and points in the opposite direction as v. v.

For the following exercises, find a unit vector in the same direction as the given vector.

22.

a = 3i + 4j

23.

b = −2i + 5j

24.

c = 10ij

25.

d= 1 3 i+ 5 2 j d= 1 3 i+ 5 2 j

26.

u = 100i + 200j

27.

u = −14i + 2j

For the following exercises, find the magnitude and direction of the vector, 0θ<2π. 0θ<2π.

28.

0,4 0,4

29.

6,5 6,5

30.

2,−5 2,−5

31.

−4,−6 −4,−6

32.

Given u = 3i − 4j and v = −2i + 3j, calculate uv. uv.

33.

Given u = −ij and v = i + 5j, calculate uv. uv.

34.

Given u= 2,4 u= 2,4 and v= 3,1 , v= 3,1 ,calculate uv. uv.

35.

Given u = 1,6 = 1,6 and v = 6,1 , = 6,1 , calculate uv. uv.

Graphical

For the following exercises, given v, v,draw v, v, 3v and 1 2 v. 1 2 v.

36.

2,−1 2,−1

37.

−1,4 −1,4

38.

−3,−2 −3,−2

For the following exercises, use the vectors shown to sketch u + v, uv, and 2u.

39.
Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,1) and v goes to (-1,2).
40.
Plot of vectors u and v extending from the same origin point. In terms of that point, u goes to (1,2) and v goes to (1,-1).
41.
Plot of vectors u and v located head to tail. Take u's start point as the origin. In terms of that, u goes from the origin to (3,-2), and v goes from (3,-2) to (2,-3)

For the following exercises, use the vectors shown to sketch 2u + v.

42.
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (3,1) and v goes from the origin to (2,-2).
43.
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,-2) and v goes from the origin to (-3,-2).

For the following exercises, use the vectors shown to sketch u − 3v.

44.
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (-4,0) and v goes from the origin to (1,-1).
45.
Plot of the vectors u and v extending from the same point. Taking that base point as the origin, u goes from the origin to (1,2) and v goes from the origin to (-2,1).

For the following exercises, write the vector shown in component form.

46.
Vector going from the origin to (-4,2).
47.
Insert figure(table) alt text: Vector going from the origin to (4,1).
48.

Given initial point P 1 =( 2,1 ) P 1 =( 2,1 )and terminal point P 2 =( 1,2 ), P 2 =( 1,2 ),write the vector v vin terms of i iand j, j,then draw the vector on the graph.

49.

Given initial point P 1 =( 4,1 ) P 1 =( 4,1 )and terminal point P 2 =( 3,2 ), P 2 =( 3,2 ),write the vector v vin terms of i iand j. j. Draw the points and the vector on the graph.

50.

Given initial point P 1 =( 3,3 ) P 1 =( 3,3 )and terminal point P 2 =( 3,3 ), P 2 =( 3,3 ),write the vector v vin terms of i iand j. j.Draw the points and the vector on the graph.

Extensions

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

51.

| v |=6,θ=45° | v |=6,θ=45°

52.

| v |=8,θ=220° | v |=8,θ=220°

53.

| v |=2,θ=300° | v |=2,θ=300°

54.

| v |=5,θ=135° | v |=5,θ=135°

55.

A 60-pound box is resting on a ramp that is inclined 12°. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.
56.

A 25-pound box is resting on a ramp that is inclined 8°. Rounding to the nearest tenth,

  1. Find the magnitude of the normal (perpendicular) component of the force.
  2. Find the magnitude of the component of the force that is parallel to the ramp.
57.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27° above the horizontal. Round to the nearest hundredth.

58.

Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of 127° above the horizontal. Round to the nearest hundredth.

59.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth.

60.

Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of 8° above the horizontal. Round to the nearest hundredth.

Real-World Applications

61.

A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

62.

A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?

63.

A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

64.

A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?

65.

A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?

66.

A woman starts walking from home and walks 6 miles at 40° north of east, then 2 miles at 15° east of south, then 5 miles at 30° south of west. If she walked straight home, how far would she have to walk, and in what direction?

67.

An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

68.

An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing from the northwest at 50 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

69.

An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

70.

An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

71.

As part of a video game, the point ( 5,7 ) ( 5,7 )is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of this point.

72.

As part of a video game, the point ( 7,3 ) ( 7,3 )is rotated counterclockwise about the origin through an angle of 40°. Find the new coordinates of this point.

73.

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

74.

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

75.

A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.

76.

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45° from the horizontal. What single force is the resultant force acting on the body?

77.

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135° from the horizontal, and 5 pounds acting on it directed 150° from the horizontal. What single force is the resultant force acting on the body?

78.

The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45° from the horizontal. What single force is needed to produce a state of equilibrium on the body?

79.

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

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