8 Applications of Trigonometry Copyright © 2009 Pearson Addison-Wesley

Applications of Trigonometry 8 8.1 The Law of Sines 8.2 The Law of Cosines 8.3 Vectors, Operations, and the Dot Product 8.4 Applications of Vectors 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.4 De Moivre’s Theorem; Powers and Roots of Complex Numbers 8.5 Polar Equations and Graphs 8.6 Parametric Equations, Graphs, and Applications Copyright © 2009 Pearson Addison-Wesley

Vectors, Operations, and the Dot Product 8.3 Vectors, Operations, and the Dot Product Basic Terminology ▪ Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ Dot Product and the Angle Between Vectors Copyright © 2009 Pearson Addison-Wesley 1.1-3

Basic Terminology Scalar: The magnitude of a quantity. It can be represented by a real number. A vector in the plane is a directed line segment. Consider vector OP O is called the initial point P is called the terminal point Copyright © 2009 Pearson Addison-Wesley

Basic Terminology Magnitude: length of a vector, expressed as |OP| Two vectors are equal if and only if they have the same magnitude and same direction. Vectors OP and PO have the same magnitude, but opposite directions. |OP| = |PO| Copyright © 2009 Pearson Addison-Wesley

Basic Terminology A = B C = D A ≠ E A ≠ F Copyright © 2009 Pearson Addison-Wesley

Two ways to represent the sum of two vectors The sum of two vectors is also a vector. The vector sum A + B is called the resultant. Two ways to represent the sum of two vectors Copyright © 2009 Pearson Addison-Wesley

Sum of Two Vectors The sum of a vector v and its opposite –v has magnitude 0 and is called the zero vector. To subtract vector B from vector A, find the vector sum A + (–B). Copyright © 2009 Pearson Addison-Wesley

Scalar Product of a Vector The scalar product of a real number k and a vector u is the vector k ∙ u, with magnitude |k| times the magnitude of u. Copyright © 2009 Pearson Addison-Wesley

Algebraic Interpretation of Vectors A vector with its initial point at the origin is called a position vector. A position vector u with its endpoint at the point (a, b) is written Copyright © 2009 Pearson Addison-Wesley

Algebraic Interpretation of Vectors The numbers a and b are the horizontal component and vertical component of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector. Copyright © 2009 Pearson Addison-Wesley

Magnitude and Direction Angle of a Vector a, b The magnitude (length) of a vector u = a, b is given by The direction angle θ satisfies where a ≠ 0. Copyright © 2009 Pearson Addison-Wesley 1.1-12

Find the magnitude and direction angle for u = 3, –2. Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE Find the magnitude and direction angle for u = 3, –2. Magnitude: Direction angle: Copyright © 2009 Pearson Addison-Wesley 1.1-13

Graphing calculator solution: Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE (continued) Graphing calculator solution: Copyright © 2009 Pearson Addison-Wesley 1.1-14

Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle θ are given by or Copyright © 2009 Pearson Addison-Wesley 1.1-15

Horizontal component: 18.7 Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS Vector w has magnitude 25.0 and direction angle 41.7°. Find the horizontal and vertical components. Horizontal component: 18.7 Vertical component: 16.6 Copyright © 2009 Pearson Addison-Wesley 1.1-16

Graphing calculator solution: Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS Graphing calculator solution: Copyright © 2009 Pearson Addison-Wesley 1.1-17

Write each vector in the figure in the form a, b. Example 3 WRITING VECTORS IN THE FORM a, b Write each vector in the figure in the form a, b. Copyright © 2009 Pearson Addison-Wesley 1.1-18

Properties of Parallelograms 1. A parallelogram is a quadrilateral whose opposite sides are parallel. 2. The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary. 3. The diagonals of a parallelogram bisect each other, but do not necessarily bisect the angles of the parallelogram. Copyright © 2009 Pearson Addison-Wesley 1.1-19

Use the law of cosines with ΔPSR or ΔPQR. Example 4 FINDING THE MAGNITUDE OF A RESULTANT Two forces of 15 and 22 newtons act on a point in the plane. (A newton is a unit of force that equals .225 lb.) If the angle between the forces is 100°, find the magnitude of the resultant vector. The angles of the parallelogram adjacent to P measure 80° because the adjacent angles of a parallelogram are supplementary. Use the law of cosines with ΔPSR or ΔPQR. Copyright © 2009 Pearson Addison-Wesley 1.1-20

The magnitude of the resultant vector is about 24 newtons. Example 4 FINDING THE MAGNITUDE OF A RESULTANT (continued) The magnitude of the resultant vector is about 24 newtons. Copyright © 2009 Pearson Addison-Wesley 1.1-21

d Vector Operations For any real numbers a, b, c, d, and k, Copyright © 2009 Pearson Addison-Wesley 1.1-22

Let u = –2, 1 and v = 4, 3. Find the following. Example 5 PERFORMING VECTOR OPERATIONS Let u = –2, 1 and v = 4, 3. Find the following. (a) u + v = –2, 1 + 4, 3 = –2 + 4, 1 + 3 = 2, 4 (b) –2u = –2 ∙ –2, 1 = –2(–2), –2(1) = 4, –2 (c) 4u – 3v = 4 ∙ –2, 1 – 3 ∙ 4, 3 = –8, 4 –12, 9 = –8 – 12, 4 – 9 = –20,–5 Copyright © 2009 Pearson Addison-Wesley 1.1-23

Unit Vectors A unit vector is a vector that has magnitude 1. j = 0, 1 Copyright © 2009 Pearson Addison-Wesley

Unit Vectors Any vector a, b can be expressed in the form ai + bj using the unit vectors i and j. Copyright © 2009 Pearson Addison-Wesley

Dot Product The dot product (or inner product) of the two vectors u = a, b and v = c, d is denoted u ∙ v, read “u dot v,” and is given by u ∙ v = ac + bd. Copyright © 2009 Pearson Addison-Wesley 1.1-26

Example 6 Find each dot product. = 2(4) + 3(–1) = 5 FINDING DOT PRODUCTS Find each dot product. (a) 2, 3 ∙ 4, –1 = 2(4) + 3(–1) = 5 (b) 6, 4 ∙ –2, 3 = 6(–2) + 4(3) = 0 Copyright © 2009 Pearson Addison-Wesley 1.1-27

Properties of the Dot Product For all vectors u, v, and w and real number k, (a) u ∙ v = v ∙ u (b) u ∙ (v + w) = u ∙ v + u ∙ w (c) (u + v) ∙ w = u ∙ w + v ∙ w (d) (ku) ∙ v = k(u ∙ v) = u ∙ kv (e) 0 ∙ u = 0 (f) u ∙ u = |u|2 Copyright © 2009 Pearson Addison-Wesley 1.1-28

Geometric Interpretation of the Dot Product If θ is the angle between the two nonzero vectors u and v, where 0° ≤ θ ≤ 180°, then or Copyright © 2009 Pearson Addison-Wesley 1.1-29

Find the angle θ between the two vectors u = 3, 4 and v = 2, 1. Example 7 FINDING THE ANGLE BETWEEN TWO VECTORS Find the angle θ between the two vectors u = 3, 4 and v = 2, 1. Copyright © 2009 Pearson Addison-Wesley 1.1-30

Dot Products For angles θ between 0° and 180°, cos θ is positive, 0, or negative when θ is less than, equal to, or greater than 90°, respectively. Copyright © 2009 Pearson Addison-Wesley

Note If a ∙ b = 0 for two nonzero vectors a and b, then cos θ = 0 and θ = 90°. Thus, a and b are perpendicular or orthogonal vectors. Copyright © 2009 Pearson Addison-Wesley 1.1-32