Copyright © 2017, 2013, 2009 Pearson Education, Inc. Applications of Trigonometry and Vectors Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

Algebraically Defined Vectors and the Dot Product 7.5 Algebraically Defined Vectors and the Dot Product Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ The Dot Product and the Angle between Vectors

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, respectively, of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector.

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.

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:

Graphing calculator solution: Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE (continued) Graphing calculator solution:

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

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

Graphing calculator solution: Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS Graphing calculator solution:

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.

Vector Operations Let a, b, c, d, and k represent real numbers.

Let u = –2, 1 and v = 4, 3. Find and illustrate the following. Example 4 PERFORMING VECTOR OPERATIONS Let u = –2, 1 and v = 4, 3. Find and illustrate the following. (a) u + v = –2, 1 + 4, 3 = –2 + 4, 1 + 3 = 2, 4

Let u = –2, 1 and v = 4, 3. Find and illustrate the following. Example 4 PERFORMING VECTOR OPERATIONS Let u = –2, 1 and v = 4, 3. Find and illustrate the following. (b) –2u = –2 ∙ –2, 1 = –2(–2), –2(1) = 4, –2

Let u = –2, 1 and v = 4, 3. Find and illustrate the following. Example 4 PERFORMING VECTOR OPERATIONS Let u = –2, 1 and v = 4, 3. Find and illustrate the following. (c) 3u – 2v = 3 ∙ –2, 1 – 2 ∙ 4, 3 = –6, 3 –8, 6 = –6 – 8, 3 – 6 = –14, –3

Unit Vectors A unit vector is a vector that has magnitude 1. j = 0, 1

Unit Vectors Any vector a, b can be expressed in the form ai + bj using the unit vectors i and j.

v = ai + bj, where i = 1, 0 and j = 0, 1. i, j Form for Vectors If v = a, b, then v = ai + bj, where i = 1, 0 and j = 0, 1.

Dot Product The dot product of the two vectors u = a, b and v = c, d is denoted u ∙ v, read “u dot v,” and is given by the following. u ∙ v = ac + bd

Example 5 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

Properties of the Dot Product For all vectors u, v, and w and real numbers k, the following hold. (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

Geometric Interpretation of the Dot Product If θ is the angle between the two nonzero vectors u and v, where 0° ≤ θ ≤ 180°, then the following holds.

Find the angle θ between the two vectors u = 3, 4 and v = 2, 1. Example 6(a) FINDING THE ANGLE BETWEEN TWO VECTORS Find the angle θ between the two vectors u = 3, 4 and v = 2, 1.

Find the angle θ between the two vectors u = 2, –6 and v = 6, 2. Example 6(b) FINDING THE ANGLE BETWEEN TWO VECTORS Find the angle θ between the two vectors u = 2, –6 and v = 6, 2.

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.

Note If u ∙ v = 0 for two nonzero vectors u and v, then cos θ = 0 and θ = 90°. Thus, u and v are perpendicular or orthogonal vectors.