1. Vectors and Applications BOMLA LacyMath Summer 2015

2. Vectors and Applications  This unit covers the topic of vectors, which loosely falls in line with the topic of right triangle trigonometry. These topics are important in many real-world applications, such as calculating the path of the wind on an airplane’s path.

3. Vectors and Applications  Vectors are used to represent quantities such as force and velocity.  Vectors have both magnitude and direction.

4. Vectors and Applications  Vectors are represented in the coordinate plane using an arrow. These arrows are drawn using an initial point and a terminal point.  Initial point: Where the vector begins  Terminal point: Where the vector ends

5. Vectors and Applications  Example 1 Draw a vector with initial point (3, 4) and terminal point (7, 8).

6. Vectors and Applications  Example 2 Draw a vector with initial point (-6, 3) and terminal point (4, -2).

7. Vectors and Applications  Example 3 Draw a vector with initial point (-1, -5) and terminal point (0, 3).

8. Vectors and Applications  Any vector that has an initial point at the origin is said to be in standard position.  A two-dimensional vector v is an ordered pair of real numbers, and is represented by its component form given by. “a” is the horizontal component “b” is the vertical component Vectors in component form start at the origin!

9. Vectors and Applications  To put a vector in component form, use the “Head Minus Tail” method. (Terminal Point) – (Initial Point)  Initial Point:(x 1, y 1 )  Terminal Point:(x 2, y 2 )

10. Vectors and Applications  Back to Example 1 (put in component form) Draw a vector with initial point (3, 4) and terminal point (7, 8).

11. Vectors and Applications  Back to Example 2 (put in component form) Draw a vector with initial point (-6, 3) and terminal point (4, -2).

12. Vectors and Applications  Back to Example 3 (put in component form) Draw a vector with initial point (-1, -5) and terminal point (0, 3).

13. Vectors and Applications  The magnitude of a vector is the length of the arrow.  The direction is the direction in which the arrow is pointing.

14. Vectors and Applications  Back to Example 1 Find the magnitude of the vector.

15. Vectors and Applications  Back to Example 2 Find the magnitude of the vector.

16. Vectors and Applications  Back to Example 3 Find the magnitude of the vector.

17. Vectors and Applications

18.  What about direction?  The direction of the vector is the angle that the vector makes with the positive x-axis.  But how do we find it?

19. Vectors and Applications  What about direction?  What are we given when we know the component form of a vector? (Think, where does the vector start?)

20. Vectors and Applications

21.  Back to Example 1 Find the direction of the vector.

22. Vectors and Applications  Back to Example 2 Find the direction of the vector.

23. Vectors and Applications  Back to Example 3 Find the direction of the vector.

24. Vectors and Applications

25.  Standard Unit Vectors These are the simplest forms of horizontal and vertical vectors that can be created. They are represented by i and j. Any vector can be written as a combination of these vectors. i = j =

26. Vectors and Applications  Back to Example 1 Write this vector as a combination of standard unit vectors.

27. Vectors and Applications  Back to Example 2 Write this vector as a combination of standard unit vectors.

28. Vectors and Applications  Back to Example 3 Write this vector as a combination of standard unit vectors.

29. Vectors and Applications  Vector Operations  Vectors can be added and subtracted by using their components.

30. Vectors and Applications  Vector Operations  Example 1 v = and u = Find u + v, u – v, and 3 u + 2 v.

31. Vectors and Applications  CW Pg. 511 #5 – 8, 13, 15 – 17, 29, 30