Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-1 7.4 Vectors, Operations, and the Dot Product 7.5Applications of Vectors Applications

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-1

7.4 Vectors, Operations, and the Dot Product

7.5Applications of Vectors

Applications of Trigonometry and Vectors7


Vectors, Operations, and the Dot Product7.4Basic Terminology ▪ Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ Dot Product and the Angle Between Vectors


Find the magnitude and direction angle for

7.4 Example 1 Finding Magnitude and Direction Angle

Magnitude:

Direction angle:


Vector v has magnitude 14.5 and direction angle 220°. Find the horizontal and vertical components.

7.4 Example 2 Finding Horizontal and Vertical Components

Horizontal component: –11.1

Vertical component: –9.3


Write each vector in the form a, b.

7.4 Example 3 Writing Vectors in the Form a, b

u: magnitude 8, direction angle 135°

v: magnitude 4, direction angle 270°

w: magnitude 10, direction angle 340°


Two forces of 32 and 48 newtons act on a point in the plane. If the angle between the forces is 76°, find the magnitude of the resultant vector.

7.4 Example 4 Finding the Magnitude of a Resultant

because the adjacent angles of a parallelogram are supplementary.

Law of cosines

Find squareroot.


Find each dot product.

7.4 Example 6 Finding Dot Products


Find the angle θ between the two vectors u = 5, –12 and v = 4, 3.

7.4 Example 7 Finding the Angle Between Two Vectors


Applications of Vectors7.5The Equilibrant ▪ Incline Applications ▪ Navigation Applications


Find the force required to keep a 2500-lb car parked on a hill that makes a 12° angle with the horizontal.

7.5 Example 2 Finding a Required Force

The vertical force BA represents the force of gravity.

BA = BC + (–AC)


7.5 Example 2 Finding a Required Force (cont.)

Vector BC represents the force with which the weight pushes against the hill.

Vector BF represents the force that would pull the car up the hill.

Since vectors BF and AC are equal, gives the magnitude of the required force.



Vectors BF and AC are parallel, so the measure of angle EBD equals the measure of angle A.

Since angle BDE and angle C are right angles, triangles CBA and DEB have two corresponding angles that are equal and, thus, are similar triangles.

Therefore, the measure of angle ABC equals the measure of angle E, which is 12°.



From right triangle ABC,

A force of approximately 520 lb will keep the car parked on the hill.


A force of 18.0 lb is required to hold a 74.0-lb crate on a ramp. What angle does the ramp make with the horizontal?

7.5 Example 3 Finding an Incline Angle

Vector BF represents the force required to hold the crate on the incline.

In right triangle ABC, the measure of angle B equals θ, the magnitude of vector BA represents the weight of the crate, and vector AC equals vector BF.


7.5 Example 3 Finding an Incline Angle (cont.)

The ramp makes an angle of about 14.1° with the horizontal.


A ship leaves port on a bearing of 25.0° and travels 61.4 km. The ship then turns due east and travels 84.6 km. How far is the ship from port? What is its bearing from port?

7.5 Example 4 Applying Vectors to a Navigation Problem

Vectors PA and AE represent the ship’s path. We are seeking the magnitude and bearing of PE.


Triangle PNA is a right triangle, so the measure of angle NAP = 90° − 25.0° = 65.0°.

7.5 Example 4 Applying Vectors to a Navigation Problem (cont.)

The measure of angle PAE = 180° − 65.0 = 115.0°.

Law of cosines

The ship is about 123.8 km from port.


To find the bearing of the ship from port, first find the measure of angle APE.


Law of sines

Now add 38.3° to 25.0° to find that the bearing is 63.3°.


A plane with an airspeed of 355 mph is headed on a bearing of 62°. A west wind is blowing (from west to east) at 28.5 mph. Find the groundspeed and the actual bearing of the plane.

7.5 Example 5 Applying Vectors to a Navigation Problem

The groundspeed is represented by |x|.



Law of cosines

The plane’s groundspeed is about 380 mph.



The bearing is about 62° + 2° = 64°.

Use the law of sines to find α, and then determine the bearing, 62° + α.


8.5 (part I) Polar Coordinates

8.1 Complex Numbers

8.2 Trigonometric (Polar) Form of Complex Numbers

8.3 The Product and Quotient Theorems

8.4 De Moivre’s Theorem; Powers and Roots of Complex Numbers

8.5 (part II) Polar Equations and Graphs

Complex Numbers, Polar Equations, and Parametric Equations8


Polar Coordinates (part I)8.5Polar Coordinate System ▪ Converting Polar and Rectangular Coordinates


Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point.

8.5 Example 1 Plotting Points With Polar Coordinates

The rectangular coordinates

of P(4, 135°) are


8.5 Example 1 Plotting Points With Polar Coordinates (cont.)

The rectangular coordinates of


8.5 Example 1 Plotting Points With Polar Coordinates (cont.)

The rectangular coordinates of are (0, –2).


Give three other pairs of polar coordinates for the point P(5, –110°).

8.5 Example 2(a) Giving Alternative Forms for Coordinates of a Point

Three pairs of polar coordinates for the point P(5, −110º) are (5, 250º), (−5, 70º), and (−5, −290º).

Other answers are possible.


Give two pairs of polar coordinates for the point with the rectangular coordinates

8.5 Example 2(b) Giving Alternative Forms for Coordinates of a Point

The point lies in quadrant II.

Since , one possible value for θ is 300°.

Other answers are possible.

Two pairs of polar coordinates are (12, 300°) and (−12, 120°).


Complex Numbers8.1Basic Concepts of Complex Numbers ▪ Complex Solutions of Equations ▪ Operations on Complex Numbers


Write as the product of real number and i, using the definition of

8.1 Example 1 Writing √–a as i√a


Solve each equation.

8.1 Example 2 Solving Quadratic Equations for Complex Solutions


8.1 Example 3 Solving a Quadratic Equation for Complex Solutions

Write the equation in standard form,then solve using the quadratic formula with a = 2, b = –2, and c = 5.


Multiply or divide as indicated. Simplify each answer.

8.1 Example 4 Finding Products and Quotients Involving Negative Radicands


Write in standard form.

8.1 Example 5 Simplifying a Quotient Involving a Negative Radicand


8.1 Example 6 Adding and Subtracting Complex Numbers

Find each sum or difference.

(a) (4 – 5i) + (–5 + 8i)

= –1 + 3i

= [4 + (–5)] + (–5i + 8i)

(b) (–6 + 3i) + (12 – 9i) = 6 – 6i

(c) (–10 + 7i) – (5 – 3i)

= –15 + 10i

(d) (15 – 8i) – (–10 + 4i) + (–25 + 12i)

= 0 + 0i

= (–10 – 5) + [7i + (3i)]

= [15 – (–10) + (–25)] + [–8i – 4i + 12i]


8.1 Example 7 Multiplying Complex Numbers

Find each product.

(a) (5 + 3i)(2 – 7i)

(b) (4 – 5i)2

= 5(2) + (5)(–7i) + (3i)(2) + (3i)(–7i)

= 10 – 35i + 6i – 21i2

= 10 – 29i – 21(–1)

= 31 – 29i

= 42 – 2(4)(5i) + (5i)2

= 16 – 40i + 25i2

= 16 – 40i + 25(–1)

= –9 – 40i


8.1 Example 7 Multiplying Complex Numbers (cont.)

(d) (9 – 8i)(9 + 8i) = 92 – (8i)2

= 81 – 64i2

= 81 – 64(–1)

= 81 + 64

= 145 or 145 + 0i

(c) (3 – i)(–3 + i) = –9 + 3i + 3i – i2

= –9 + 6i – (–1)

= –9 + 6i + 1= –8 + 6i


8.1 Example 8 Simplifying Powers of i

Simplify each power of i.

Write the given power as a product involving or .

(a) (b)

(a) (b)


8.1 Example 8 Simplifying Powers of i (cont.)

(c)


8.1 Example 9(a) Dividing Complex Numbers

Write the quotient in standard form a + bi.

Multiply the numerator and denominator by the complex conjugate of the denominator.

i2 = –1

Combine terms.

Lowest terms; standard form

Multiply.


8.1 Example 9(b) Dividing Complex Numbers

Write the quotient in standard form a + bi.

Multiply the numerator and denominator by the complex conjugate of the denominator.

–i2 = 1

Lowest terms; standard form

Multiply.


Trigonometric (Polar) Form of Complex Numbers8.2The Complex Plane and Vector Representation ▪ Trigonometric (Polar) Form ▪ Converting Between Rectangular and Trigonometric (Polar) Forms ▪ An Application of Complex Numbers to Fractals


Find the sum of 2 + 3i and –4 + 2i. Graph both complex numbers and their resultant.

8.2 Example 1 Expressing the Sum of Complex Numbers Graphically

(2 + 3i) + (–4 + 2i) = –2 + 5i


Express 10(cos 135° + i sin 135°) in rectangular form.

8.2 Example 2 Converting From Trigonometric Form to Rectangular Form


Write 8 – 8i in trigonometric form.

8.2 Example 3(a) Converting From Rectangular Form to Trigonometric Form

The reference angle for θ is 45°. The graph shows that θ is in quadrant IV, so θ = 315°.


Write –15 in trigonometric form.

8.2 Example 3(b) Converting From Rectangular Form to Trigonometric Form

–15 + 0i is on the negative x-axis, so θ = 180°.

–15 = –15 + 0i


Write each complex number in its alternative form, using calculator approximations as necessary.

8.2 Example 4 Converting Between Trigonometric and Rectangular Forms Using Calculator Approximations

(a) 7(cos 205° + i sin 205°) ≈ –6.3442 – 2.9583i


8.2 Example 4 Converting Between Trigonometric and Rectangular Forms Using Calculator Approximations (cont.)

(b) –7 + 2i

x = −7 and y = 2

The reference angle for θ is approximately 15.95°. The graph shows that θ is in quadrant II, so θ = 180° – 15.95° = 164.05°.


The Product and Quotient Theorems8.3Products of Complex Numbers in Trigonometric Form ▪ Quotients of Complex Numbers in Trigonometric Form


8.3 Product Theorem

1 1 1 2 2 2(cos sin ) (cos sin )r i r i

1 2 1 2 1 2[cos( ) sin( )]r r i

Can be written as


Find the product of 4(cos 120° + i sin 120°) and5(cos 30° + i sin 30°). Write the result in rectangular form.

8.3 Example 1 Using the Product Theorem


8.3 Quotient Theorem

1 1 1

2 2 2

(cos sin )

(cos sin )

r i

r i

11 2 1 2

2

[cos( ) sin( )]r

ir

Can be written as


Find the quotient Write the result in rectangular form.

8.3 Example 2 Using the Quotient Theorem


Use a calculator to find the following. Write the results in rectangular form.

8.3 Example 3 Using the Product and Quotient Theorems With a Calculator


8.3 Example 3 Using the Product and Quotient Theorems With a Calculator (cont.)


De Moivre’s Theorem; Powers and Roots of Complex Numbers8.4Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of Complex Numbers


Find and express the result in rectangular form.

8.4 Example 1 Finding a Power of a Complex Number

First write in trigonometric form.

and

Because x and y are both positive, θ is in quadrant I, so θ = 45°.


8.4 Example 1 Finding a Power of a Complex Number (cont.)


Find the three cube roots of 8 (cos 180o + i sin 180o). Write the roots in rectangular form.

8.4 Example 2 Finding Complex Roots

Note that [2 (cos 60o + i sin 60o)]3 = 8 (cos 180o + i sin 180o).

So one cube root is 2 (cos 60o + i sin 60o)

The other 2 are apart 360

1203

2 (cos 180o + i sin 180o) and 2 (cos 300o + i sin 300o)

If cubed all 3 will yield 8 in rectangular form


Find all fourth roots of Write the roots in rectangular form.

8.4 Example 3 Finding Complex Roots

First write in trigonometric form.

Because x and y are both negative, θ is in quadrant III, so θ = 240°.


8.4 Example 3 Finding Complex Roots (cont.)


Polar Equations and Graphs (part II)8.5Graphs of Polar Equations ▪ Converting from Polar to Rectangular Equations ▪ Classifying Polar Equations


8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles

For each rectangular equation, give the equivalent polar equation and sketch its graph.

(a) y = 2x – 4

In standard form, the equation is 2x – y = 4, so a = 2, b = –1, and c = 4.

The general form for the polar equation of a line is

y = 2x – 4 is equivalent to


8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (cont.)



This is the graph of a circle with center at the origin and radius 5.

Note that in polar coordinates it is possible for r < 0.




8.5 Example 8 Converting a Polar Equation to a Rectangular Equation

#53 2sin( )r

#582

4cos sinr

#60 5csc( )r


8.5 Example 4 Graphing a Polar Equation (Cardioid)

Find some ordered pairs to determine a pattern of values of r.


8.5 Example 4 Graphing a Polar Equation (Cardioid) (cont.)

Connect the points in order from (1, 0°) to (.5, 30°) to (.1, 60°) and so on.


8.5 Example 5 Graphing a Polar Equation (Rose)

Find some ordered pairs to determine a pattern of values of r.


8.5 Example 5 Graphing a Polar Equation (Rose) (cont.)

Connect the points in order from (4, 0°) to (3.6, 10°) to (2.0, 20°) and so on.


8.5 Example 7 Graphing a Polar Equation (Spiral of Archimedes)

Graph r = –θ (θ measured in radians).

Go to wzgrapher. Use domain [0, 20π]

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Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 7-1 7.4 Vectors, Operations, and the Dot Product 7.5Applications of Vectors Applications