1. Tuesday, March 3, 2015
HW: pg even on problems you also need to do the operations geometrically. Do Now:
Take out your pencil, notebook, and calculator.
Point P is on the terminal side of the angle x. Find the measure of x.
P(5,9)
Objectives:
•You will be able to apply the arithmetic of vectors to solve real world problems.
Agenda:
Do Now
Vectors introduction (6.1)

2. 6.1
Vectors in the Plane

3. What you’ll learn about
How to represent vectors as directed line segments
How to perform basic Vector Operations
How to write vectors as linear combinations of Unit Vectors
How to find the Direction Angles of vectors
How to use vectors to model and solve real-life problems
… and why
These topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.

4. 6.1 Vectors in the Plane
Many quantities in geometry and physics, such as area, time,
and temperature, can be represented by a single real number.
Other quantities, such as force and velocity, involve both
magnitude and direction and cannot be completely
characterized by a single real number.
To represent such a quantity, we use a directed line segment.
The directed line segment 𝑃𝑄 has initial point P and terminal
point Q and we denote its magnitude (length) by Q PQ
Terminal Point P
Initial Point

5. Directed Line Segment
Vector notation

6. Two-Dimensional Vector
A 2D vector v is an ordered pair of real numbers in component form 𝑎, 𝑏 .
Magnitude of v is the length or the arrow from the origin to the point (a, b)
Direction of v is the direction in which the arrow is pointing.
The vector 0= 0, 0 , called the zero vector, has length 0 and direction of 0.

7. Associating arrows with the vectors they represent
Head Minus Tail (HMT) Rule
If an arrow has an initial point ( 𝑥 1 , 𝑦 1 ) 𝑎𝑛𝑑 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡( 𝑥 2 , 𝑦 2 ), it represents the vector 𝑥 2 − 𝑥 1 , 𝑦 2 − 𝑦 1

8. Showing vectors are equivalent.
Any 2 arrows with the same length and pointing in the same direction represent the same vector.
Example 1: Show that the arrow from R=(-4,2) to S=(-1, 6) is equal to the arrow from P=(3,4) to Q=(0,0)
−1− −4 , 6−2 = 3,4
3−0, 4−0 = 3,4
Therefore, the arrows represent
The same vector even though they
Have different initial and terminal points

9. Vector Archery –exploration 1
With your partner complete the vector archery exploration in your notes on page 504.
We will discuss in 10 minutes.

10. Magnitude Magnitude – called absolute value of v
Why? What is magnitude in vectors?
Denoted by 𝑣 𝑜𝑟 𝑣

11. Magnitude
Comes from distance formula!

12. Example Finding Magnitude of a Vector

13. Example Finding Magnitude of a Vector

14. Vector Addition and Scalar Multiplication

16. Example Performing Vector Operations

17. Example Performing Vector Operations

18. Unit Vectors

20. Example Finding a Unit Vector

21. Example Finding a Unit Vector

22. Standard Unit Vectors

23. Resolving the Vector

25. Example Finding the Components of a Vector

26. Example Finding the Components of a Vector

30. Example Finding the Direction Angle of a Vector

31. Example Finding the Direction Angle of a Vector

32. Velocity and Speed The velocity of a moving object is a vector
because velocity has both magnitude and
direction. The magnitude of velocity is speed.