1. Vectors

2. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –E.g. distance, mass, speed, temperature, etc. A vector is a set of numbers that describe both a magnitude and direction –E.g. velocity (the magnitude of velocity is speed), force, momentum, etc. Notation: a vector-valued variable is differentiated from a scalar one by using bold or the following symbol: A

3. 3 Characteristics of Vectors A Vector is something that has two and only two defining characteristics: 1.Magnitude: the 'size' or 'quantity' 2.Direction: the vector is directed from one place to another.

4. 4 Direction Speed vs. Velocity Speed is a scalar, (magnitude no direction) - such as 5 feet per second. Speed does not tell the direction the object is moving. All that we know from the speed is the magnitude of the movement. Velocity, is a vector (both magnitude and direction) – such as 5 ft/s Eastward. It tells you the magnitude of the movement, 5 ft/s, as well as the direction which is Eastward.

5. Which of the following *is* a vector? A.Direction B.Speed C.Velocity

6. 6 Graphing Vectors Note: moving a vector does not change it. A vector is only defined by its magnitude and direction, not starting location Examples Vectors with the same direction

7. 7 Expressing Vectors as Ordered Pairs How can we express this vector as an ordered pair? Use Trigonometry

8. 8

9. 9 Now Let’s Try Express this vector as an ordered pair.

10. 10 Now You Try Express this vector as an ordered pair. 70° 44

11. What is the component form of the vector with magnitude 44 and an angle of 70 ° with the horizontal? A. B. C. D.

12. 12 Adding Vectors Add vectors A and B

13. 13 Adding Vectors On a graph, add vectors using the “head-to-tail” rule: Move B so that the head of A touches the tail of B Note: “moving” B does not change it. A vector is only defined by its magnitude and direction, not starting location.

14. 14 Adding Vectors The vector starting at the tail of A and ending at the head of B is C, the sum (or resultant) of A and B.

15. 15 Adding Vectors Let’s go back to our example: Now our vectors have values.

16. 16 Adding Vectors What is the value of our resultant? GeoGebra Investigation Practice

17. Vector Operations with Coordinates  Vector Addition  v + u =  Vector Subtraction  v - u =  Scalar Multiplication  kv =

18. Examples: If u = and v =, find: u + v u – v 2u – 3v

19. If u = and v =, what is 3v – 2u? A. B. C. D. [Default] [MC Any] [MC All]

20. Component Form You can subtract the coordinates. IT IS ALWAYS B – A! The magnitude of vector AB is found using the distance formula: (x 1,y 1 ) (x 2,y 2 ) (x 2 – x 1 ) (y 2 – y 1 )

21. Example Given A(4, 2) and B(9, -1), express in component form. Find and

22. Given A(-2, 6) and B(1, 10), express in component form. A. B. C. D. [Default] [MC Any] [MC All]

23. Given A(-2, 6) and B(1, 10), what is ? A.5 B.√17

24. Unit Vectors A unit vector is a vector of length 1. They are used to specify a direction. To find a unit vector, u, in an arbitrary direction, for example, in the direction of vector a, where a=, divide the vector by its magnitude (this process is called normalization). If a=, then is a unit vector in the same direction as a.

25. What is the unit vector in the direction of v = A. B. C. D. [Default] [MC Any] [MC All]

26. What is the unit vector in the direction of v = ? A. B. C. D.