1. Chapter 6 Additional Topics in Trigonometry
Pre-Calculus
Chapter 6
Additional Topics in Trigonometry

2. 6.3 Vectors in the Plane Objectives:
Represent vectors as directed line segments.
Write the component form of vectors.
Perform basic vector operations and represent vectors graphically.
Write vectors as linear combinations of unit vectors.
Find the direction angles of vectors.
Use vectors to model & solve real-life problems.

3. Question….
How do we represent quantities that have both a size (length or magnitude) and a direction?

4. Answer: Vectors
A vector is a quantity with both a size (magnitude) and a direction.
(Note: A quantity with size only is called a scalar.)

5. displacement, velocity, force
For Example….
A ball flies through the air at a certain speed and in a particular direction.
The speed and direction are the velocity of the ball.
The velocity is a vector quantity since it has both a magnitude and a direction.
Examples of vector quantities:
displacement, velocity, force

6. How Do We Represent Vectors?
A vector is represented by a directed line segment PQ with initial point P and terminal point Q.
The arrow defines the direction.
Direction is found using Slope Formula.
The length of the segment defines the magnitude.
Magnitude is found using Distance Formula. Q P

7. Vector Notation Vectors can be represented in different ways:
Lower-case, boldface letters such as u, v, and w.
The letters of the endpoints of the directed line segment with an arrow above.
Magnitude is denoted by

8. Equivalent Vectors Equivalent Vectors:
Have the same direction (parallel).
Have the same magnitude. v u

9. Example 1
Let u be represented by the directed line segment from P = (0, 0) to Q = (3, 2), and let v be represented by the directed line segment from R = (1, 2) to S = (4, 4). Show that u = v.
Magnitude = √13. Slope = 2/3.

10. Results for Example 1

11. Vector in Standard Positionx y
(u1, u2)
Standard Position:
Initial point is the origin (0, 0).
Represented uniquely by its terminal point (u1, u2).
If initial point is not the origin:
Can be rewritten as a vector in standard position.
This is called the component form of the vector. x y
P (p1, p2)
Q (q1, q2)

12. Component Form of a Vector
The component form of the vector with
Initial Point P = (x1, y1) and
Terminal Point Q = (x2, y2) is given by:
The magnitude (or length) of v is given by:

13. Zero Vector and Unit Vector
Both the initial point and the terminal point lie at the origin.
Notation:
Unit Vector:
Magnitude of the vector is equal to 1.

14. Example 2
Find the component form and magnitude of the vector v that has initial point (4, –7) and terminal point (–1, 5).
v = <– 1 – 4, 5 – (– 7)> = < – 5, 12>. Magnitude = 13.

15. Equal Vectors
Two vectors u = <x1, y1> and v = <x2, y2> are equal if and only if x1 = x2 and y1 = y2.
Example 1 (again):
Vector u from P = (0, 0) to Q = (3, 2), and vector v from R = (1, 2) to S = (4, 4). Show that u = v by writing each in component form.
u = <3-0, 2-0> = <3, 2>. v = <4-1, 4-2> = <3, 2>

16. Vector Operations
Let u = (x1, y1), v = (x2, y2), and let k be a scalar.
Scalar Multiplication ku = (kx1, ky1)
Vector Addition u + v = (x1+x2, y1+ y2)
Vector Subtraction u  v = (x1  x2, y1  y2)

17. Scalar Multiplication
The product of a vector and a constant, k (“scalar”).
New vector is │k│ times as long as v.
If k is positive, kv has same direction as v.
If k is negative, kv has opposite direction of v.

18. Vector Addition To add vectors u and v:
Place the initial point of v at the terminal point of u.
Draw the vector with the same initial point as u and the same terminal point as v. u v v u
u + v

19. Diagram for Vector Addition
“Parallelogram Law” – The sum is the “resultant” of the parallelogram.

20. Vector Subtraction To subtract vectors u and v:
Place the initial point of v at the initial point of u.
Draw the vector u – v from the terminal point of v to the terminal point of u. u v u v
u  v

21. Definition of Vector Addition and Scalar Multiplication
Let u = <u1, u2> and v = <v1, v2> be vectors and let k be a scalar.
The sum of u and v is the vector
The scalar multiple of k times u is the vector

22. Example 3 Let v = <–2, 5> and w = <3, 4>.
Find each of the following vectors. 2v
w – v
v + 2w
a. <–4, 10>. b. <5, –1>. c. <4, 13>.

23. Graphical Representation of Solutions

24. Properties of Vector Addition and Scalar Multiplication
Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true.
u + v = v + u
(c + d)u = cu + du
(u + v) + w = u + (v + w)
c(u + v) = cu + cv
u + 0 = u
1(u) = u
u + (–u) = 0
0(u) = 0
c(du) = (cd)u
|| cv || = | c | * || v ||

25. The Unit Vector
A vector, u, with magnitude equal to 1 that has the same direction as the given vector, v.
To find u, divide v by its length.
Note that u is a scalar multiple of v.
The vector u is called a unit vector in the direction of v.

26. Example 4
Find a unit vector in the direction of v = <–2, 5> and verify that the result has a magnitude of 1.
<– 2/√29, 5/√29>

27. Standard Unit Vectors The unit vectors
i = <1, 0> and j = <0, 1>
are called the standard unit vectors.
Note that the standard unit vector i is not the imaginary number

28. Linear Combination Vector v = <v1, v2> can be written as
The scalars v1 and v2 are called the horizontal and vertical components of v.
The vector sum v1i + v2j is called the linear combination of the standard unit vectors i and j.

29. Example 5
Let u be the vector with initial point (2, –5) and terminal point (–1, 3). Write u as a linear combination of the standard unit vectors i and j.
Write component form <–3, 8> first. Then write as –3i + 8j.

30. Example 6 Let u = –3i + 8j and v = 2i – j.
Find 2u – 3v without converting the vectors to component form.
– 12i + 19j

31. Direction Angles
The direction angle  of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies.
If , then x y v θ x y θ v x y v
(x, y)

32. Example 7 Find the direction angle of each vector:
tan θ = 1, θ = 45°. b. Vector in QIV. tan θ = – 4/3, θ’ = 53.13°, θ = °.

33. Example 8
Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle of 30° below the horizontal, as shown.
Mag of v = 100, θ = 210°. v = 100 cos 210° i sin 210° j = <– 50√3, – 50>. Verify mag.

34. Homework 6.3
Worksheet 6.3
<– 2/√29, 5/√29>