1. 11. Section 12.1 Vectors

2. 12.1 Vectors What is a vector and how do you combine them?

3. Vectors Quantities that we measure that have magnitude but not direction are called scalars. Quantities such as force, displacement or velocity that have direction as well as magnitude are called vectors and are represented by directed line segments. A B initial point terminal point The magnitude is

4. Vectors A vector is in standard position if the initial point is at the origin. x y The component form of this vector is:

5. Vectors Vectors are equal if they have the same magnitude (length) and direction (slope). To find the magnitude Example – Find the magnitude of

6. Example P Q (-3,4) (-5,2) The component form of is: v (-2,-2) The magnitude is

7. More Vocabulary If Then v is a unit vector. is the zero vector and has no direction. can also be written ai + bj where i is the x component and j is the y component

8. Example Find a vector that is a)parallel and b) Orthogonal (perpendicular) to f(x) = 4x 2 at (2, 16) or 1i + 16j or 16i – 1j

9. (Add the components.) (Subtract the components.) Vector Operations

10. Vector Operations: Scalar Multiplication: Negative (opposite):

11. v v u u u+v u + v is the resultant vector. (Parallelogram law of addition) Adding Vectors Graphically

12. The dot product (also called inner product) is defined as: Example: Notice the answer is scalar, not a vector

13. The angle between two vectors is given by: This comes from the law of cosines.

14. Find the angle between vectors u and v : Example:

15. A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E

16. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E u

17. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u 60 o

18. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u We need to find the magnitude and direction of the resultant vector u + v. u+v

19. N E v u The component forms of u and v are: u+v 500 70 Therefore: and:

20. N E The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5 o north of east. 538.4 6.5 o 

21. Assignment: Worksheet