copyright © 2009 pearson addison-wesley 7.4-1 7 applications of trigonometry and vectors

Copyright © 2009 Pearson Addison-Wesley 7.4-1

7Applications of Trigonometry and Vectors


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

7.5 Applications of Vectors

7Applications of Trigonometry and Vectors

Copyright © 2009 Pearson Addison-Wesley

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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


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


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|


Basic Terminology

A = B C = D A ≠ E A ≠ F


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


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).


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.


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


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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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 satisfieswhere a ≠ 0.


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Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE

Find the magnitude and direction angle for u = 3, –2.

Magnitude:

Direction angle:


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Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE (continued)

Graphing calculator solution:


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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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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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Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS

Graphing calculator solution:


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Example 3 WRITING VECTORS IN THE FORM a, b

Write each vector in the figure in the form a, b.


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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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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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Example 4 FINDING THE MAGNITUDE OF A RESULTANT (continued)

The magnitude of the resultant vector is about24 newtons.


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Vector Operations

For any real numbers a, b, c, d, and k,


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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


Unit Vectors

A unit vector is a vector that has magnitude 1.

i = 1, 0 j = 0, 1


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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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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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


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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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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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Example 7 FINDING THE ANGLE BETWEEN TWO VECTORS

Find the angle θ between the two vectors u = 3, 4 and v = 2, 1.


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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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 7.4-1 7 applications of trigonometry and vectors

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magnitude u

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finding magnitude

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