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

2. 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

3. 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

4. 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.

5. 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.

6. 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:

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

8. 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

9. 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

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

11. 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.

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

13. 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

14. 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

15. 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

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

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

18. 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.

19. 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

20. 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

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.