1. 8
Applications of Trigonometry
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2. Applications of Trigonometry8
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
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3. 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
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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
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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|
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6. Basic Terminology A = B C = D A ≠ E A ≠ F
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7. 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
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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).
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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.
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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
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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.
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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.
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13. 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:
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14. Graphing calculator solution:
Example 1
FINDING MAGNITUDE AND DIRECTION ANGLE (continued)
Graphing calculator solution:
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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
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16. 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
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17. Graphing calculator solution:
Example 2
FINDING HORIZONTAL AND VERTICAL COMPONENTS
Graphing calculator solution:
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18. 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.
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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.
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20. 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.
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21. 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.
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22. d Vector Operations For any real numbers a, b, c, d, and k,
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23. 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
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24. Unit Vectors A unit vector is a vector that has magnitude 1.
j = 0, 1
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25. Unit Vectors
Any vector a, b can be expressed in the form ai + bj using the unit vectors i and j.
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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.
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27. 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
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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
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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
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30. 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.
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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.
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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.
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