1. Copyright © 2009 Pearson Addison-Wesley 7.4-1 7 Applications of Trigonometry and Vectors

2. Copyright © 2009 Pearson Addison-Wesley 7.4-2 7.1 Oblique Triangles and the Law of Sines 7.2 The Ambiguous Case of the Law of Sines 7.3 The Law of Cosines 7.4 Vectors, Operations, and the Dot Product 7.5Applications of Vectors 7 Applications of Trigonometry and Vectors

3. Copyright © 2009 Pearson Addison-Wesley1.1-3 7.4-3 Vectors, Operations, and the Dot Product 7.4 Basic Terminology ▪ Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ Dot Product and the Angle Between Vectors

4. Copyright © 2009 Pearson Addison-Wesley 7.4-4 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

5. Copyright © 2009 Pearson Addison-Wesley 7.4-5 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|

6. Copyright © 2009 Pearson Addison-Wesley 7.4-6 Basic Terminology A = BC = D A ≠ EA ≠ F

7. Copyright © 2009 Pearson Addison-Wesley 7.4-7 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

8. Copyright © 2009 Pearson Addison-Wesley 7.4-8 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).

9. Copyright © 2009 Pearson Addison-Wesley 7.4-9 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.

10. Copyright © 2009 Pearson Addison-Wesley 7.4-10 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

11. Copyright © 2009 Pearson Addison-Wesley 7.4-11 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.

12. Copyright © 2009 Pearson Addison-Wesley1.1-12 7.4-12 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.

13. Copyright © 2009 Pearson Addison-Wesley1.1-13 7.4-13 Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE Find the magnitude and direction angle for u =  3, –2 . Magnitude: Direction angle:

14. Copyright © 2009 Pearson Addison-Wesley1.1-14 7.4-14 Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE (continued) Graphing calculator solution:

15. Copyright © 2009 Pearson Addison-Wesley1.1-15 7.4-15 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

16. Copyright © 2009 Pearson Addison-Wesley1.1-16 7.4-16 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

17. Copyright © 2009 Pearson Addison-Wesley1.1-17 7.4-17 Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS Graphing calculator solution:

18. Copyright © 2009 Pearson Addison-Wesley1.1-18 7.4-18 Example 3 WRITING VECTORS IN THE FORM  a, b  Write each vector in the figure in the form  a, b .

19. Copyright © 2009 Pearson Addison-Wesley1.1-19 7.4-19 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.

20. Copyright © 2009 Pearson Addison-Wesley1.1-20 7.4-20 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.

21. Copyright © 2009 Pearson Addison-Wesley1.1-21 7.4-21 Example 4 FINDING THE MAGNITUDE OF A RESULTANT (continued) The magnitude of the resultant vector is about 24 newtons.

22. Copyright © 2009 Pearson Addison-Wesley1.1-22 7.4-22 Vector Operations For any real numbers a, b, c, d, and k,

23. Copyright © 2009 Pearson Addison-Wesley1.1-23 7.4-23 Example 5 PERFORMING VECTOR OPERATIONS Let u =  –2, 1  and v =  4, 3 . Find the following. (a)u + v (b)–2u (c)4u – 3v =  –2, 1  +  4, 3  =  –2 + 4, 1 + 3  =  2, 4  = –2 ∙  –2, 1  =  –2(–2), –2(1)  =  4, –2  = 4 ∙  –2, 1  – 3 ∙  4, 3  =  –8, 4  –  12, 9  =  –8 – 12, 4 – 9  =  –20,–5 

24. Copyright © 2009 Pearson Addison-Wesley 7.4-24 Unit Vectors  A unit vector is a vector that has magnitude 1. i =  1, 0  j =  0, 1 

25. Copyright © 2009 Pearson Addison-Wesley 7.4-25 Unit Vectors Any vector  a, b  can be expressed in the form ai + bj using the unit vectors i and j.

26. Copyright © 2009 Pearson Addison-Wesley1.1-26 7.4-26 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.

27. Copyright © 2009 Pearson Addison-Wesley1.1-27 7.4-27 Example 6 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

28. Copyright © 2009 Pearson Addison-Wesley1.1-28 7.4-28 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

29. Copyright © 2009 Pearson Addison-Wesley1.1-29 7.4-29 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

30. Copyright © 2009 Pearson Addison-Wesley1.1-30 7.4-30 Example 7 FINDING THE ANGLE BETWEEN TWO VECTORS Find the angle θ between the two vectors u =  3, 4  and v =  2, 1 .

31. Copyright © 2009 Pearson Addison-Wesley 7.4-31 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.

32. Copyright © 2009 Pearson Addison-Wesley1.1-32 7.4-32 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.