1. Mathematical Interlude 1 Spaces, Trigonometry, and Vectors 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

2. Spatial Coordinates A spatial coordinate is an ordered tuple of one or more real numbers. (-14.2, 6.0, 23.1) Coordinates define position in a coordinate system, or space. Spaces, Trigonometry, and Vectors 2

3. 1637 by Rene Descartes Specified by three, orthogonal vectors, each of length one, and an origin. Usually shown as Cartesian Coordinate System Spaces, Trigonometry, and Vectors 3 x y z origin

4. More Coordinates The order of (x, y, z) is important. The spatial coordinates have the dimensions of length. We add a positive-going time coordinate to make a reference frame. We assume time flows uniformly, and times can be compared at different locations. Spaces, Trigonometry, and Vectors 4

5. Even More Coordinates We’ll have more to say about coordinates when we reach the topic of vectors, later in this segment. Spaces, Trigonometry, and Vectors 5

6. Trigonometry: Degrees and Radians Spaces, Trigonometry, and Vectors 6  One revolution equals 360 degrees. One revolution equals 2  radians.

7. Trigonometric Functions Spaces, Trigonometry, and Vectors 7    hypotenuse = h opposite = o adjacent = a

8. Trigonometric Functions (cont.) Spaces, Trigonometry, and Vectors 8    hypotenuse = h opposite = o adjacent = a ( soh ) ( cah ) ( toa )

9. Trigonometric Functions (cont.) Mnemonic: Chief Soh-Cah-Toa or, if you prefer, Camp Soh-Cah-Toa Spaces, Trigonometry, and Vectors 9

10. sin() and cos() Spaces, Trigonometry, and Vectors 10

11. tan() too Spaces, Trigonometry, and Vectors 11

12. Common Circular Motion Spaces, Trigonometry, and Vectors 12  r

13. More Trigonometry Spaces, Trigonometry, and Vectors 13

14. Vectors A vector is a geometric object with a length and a direction. You can imagine it as an arrow floating in space that can be moved around freely. Spaces, Trigonometry, and Vectors 14

15. Representing Vectors Spaces, Trigonometry, and Vectors 15

16. Representing Vectors (cont.) Spaces, Trigonometry, and Vectors 16

17. Vector Arithmetic (I) Spaces, Trigonometry, and Vectors 17

18. We can add two (or more) vectors together by joining them tip-to-tail to form a quadrilateral: Vector Arithmetic (I) (cont.) Spaces, Trigonometry, and Vectors 18

19. Vector Components Spaces, Trigonometry, and Vectors 19

20. Vector Components (cont.) Spaces, Trigonometry, and Vectors 20 x y z origin

21. Products of Vectors Spaces, Trigonometry, and Vectors 21

22. Dot Product of Two Vectors Spaces, Trigonometry, and Vectors 22

23. Two Useful Aspects of the Dot Product Spaces, Trigonometry, and Vectors 23

24. Next Up Calculus Spaces, Trigonometry, and Vectors 24