1. Vectors in the Plane Objectives: Define a vector What are the basic terms associated with vectors?

2. Definition A vector is a quantity that has magnitude (size) and direction. It is used to measure quantities, such as force, velocity and acceleration.

3. A vector is a displacement A vector is a translation Traveling northeast at a speed of 60 mph. 60 mph NE Example

4. Traveling west at a speed of 30 mph. 30 mph west

5. A vector is represented using a directed line segment.

6. Definition Vectors that point in the opposite direction and have the same length are opposite vectors.

7. Definition The magnitude of a vector is represented by its length. If, then the magnitude is : a b

8. Definition Vectors may be multiplied by a scaler. A scaler is a real number. It has magnitude (size) only.

9. Given is a scalar multiple of Describe the effect on the vector when:. “stretches” the vector “shrinks” the vector “reflects” the vector through a point

10. Definition Vectors are parallel if and only if they are scalar Multiples of each other.

11. Given Points A(3,2) B(6,4) C(4, -2) and D(2,-1) and Express each as a translation Do Now:

12. Every vector consists of at least 2 component vectors:1 in the x direction and 1 in the y direction This vector is in standard position. Component Vectors Aim: How do we represent vectors algebraically ?

13. Important Idea A vector may be represented by subtraction of the ordered pairs.

14. Example Find the component form of the vector v and sketch the vector with its initial point at the origin.

15. Definition Vector Addition: If and then is also called or resultant vector. the vector sum

16. Try This If and, find

17. Definition Vector Subtraction: If and, then:

18. Vector Subtraction:

19. Definition Scaler Multiplication: If and r is a real number, then:

20. Try This If and r =2, find and sketch its graph in standard position.

21. Definition and are unit vectors. Any vector can be written as a scaler times i plus a scaler times j.

22. Example

23. Try This Write using unit vectors. 5 i -3 j

24. A Force F of 10 Newtons acts at an angle of with the positive x-axis. Express F as a vector in component form.

25. Given A(0,4) and B(6,1). Find the coordinates of point P that is 2/3 of the way from A to B. NEXT DO NOW Try This

26. Solution:

27. Given A(8,4) and B(-1,7). Find the coordinates of point P that is 2/3 of the way from A to B. Another One

29. Describing Position: The position of an object at time t is given as and are standard position vectors Where is the object at t=0? t=1 t=2? t = 3? t = 4?

30. tR(t) 0 1 2 3 4

31. What is the speed of the Object? What direction is the Object moving?

32. Vector Equation of a line Given standard position vectors and The equation of a line is: is the Initial Position is the Velocity or Direction Vector is the Speed

33. Fly What is the speed of the Fly? What direction is the Fly moving How are these Flies different? Fly 2 Fly 3

34. Can this spider catch any of the Flies? Explain.

35. An object moves with constant velocity. At t = 0 it is at A(5,3) and at t = 3 it is at B(−4,15). a) find its velocity b) find its speed c) Write a vector equation describing the objects movement. Example