1. 11.5 Vectors in the Plane Do Now A man is at the top of a lighthouse 100 ft high and spots a ship in the ocean. His angle of depression to the ship is 5 degrees. How far away is the ship from the lighthouse?

2. HW Review

3. Vectors Vectors represent quantities that have both magnitude (length) and direction A two-dimensional vector v is determined by two points in the plane: an initial point P and a terminal point Q. The magnitude of v denoted ||v||, is the distance from P to Q

4. Vector notes 2 vectors are parallel if the lines through each vector are parallel A vector is translated when it is moved parallel to itself without changing length or direction 2 vectors are equivalent if they have the same length and direction

5. Vector Notation A vector V whose initial point is the origin and whose terminal point is (a, b) can be written as where a is the horizontal component and b is the vertical component of the vector

6. Component Form of a Vector The component form of a ray AC with andis We assume all vectors are based at the origin unless otherwise stated

7. Ex Determine whether the two vectors are equivalent where:

8. Ex Sketch the vector v = based at P = (1, 4) and the vector v 0 equivalent to v based at the origin.

9. Vector Addition The sum of two vectors is the vector needed to travel both vectors in order The sum is also called the resultant of the vectors

10. Vector Operations Scalar Multiplication Vector Addition Vector Subtraction Zero Vector

11. Linear Combinations Any vector u can be expressed as a linear combinations of two vectors v and w. u = av + bw Most linear combinations are made up of the unit vectors

12. Ex Express vector u = as a linear combination of v = and w =

13. Unit Vector A vector of magnitude 1 is called a unit vector If v is a vector and is not a zero vector, then a unit vector with the same direction as v is

14. Ex Find the unit vector in the direction of v =

15. Closure Sketch v =, w = and v + w HW: p.650 #5, 9 15 17 25 31 37 42 48