1. Vector Components

2. Calculator Trig Functions Make sure calculator is in DEG NOT RAD or GRAD (turn off/on)

3. Practice: find the following cos 30º sin 30º cos 60º sin 60º cos 45º sin 45º cos 23º sin 23º 0.866 0.500 0.866 0.707 0.921 0.391

4. Vector Components Any vector is the vector sum of an x-component and a y-component a = a x + a y a axax ayay y x θ

5. Vector Components Find the magnitude of the vector components a x = a cosθa y = a sinθ a axax ayay y x θ

6. Example Resolve a vector of 5 cm at 60º into it’s x and y components: a y = a sinθ a x = a cosθ = 5 sin 60º= 5 cos 60º a y = 4.33 a x = 2.5 note: sin 60º = 0.866cos60º = 0.5

7. Example Resolve 10 cm at 45º into x and y a y = a sinθ a x = a cosθ = 10 sin 45º= 10 cos 45º a y = 7.07 cm a x = 7.07 cm

8. 10 cm at 45º has positive x and y components. x -y -x y 0º0º 270º 180º 90º

9. Example Resolve a vector of 4 m/s at 143º a y = a sinθ a x = a cosθ = 4 sin 143º= 4 cos 143º a y = 2.41 m/s a x = -3.19 m/s note: sin 143º = 0.602 cos 143º = -0.799 why negative?

10. 4 m/s at 143º has a positive y but a negative x component x -y -x y 0º0º 270º 180º 90º

11. Example Resolve a vector of 92 m/s 2 at 230º a y = a sinθ a x = a cosθ = 92 sin 230º= 92 cos 230º a y = - 70.5 m/s a x = -59.1 m/s note: sin 230º = -0.766 cos 230º = -0.642

12. 92 m/s 2 at 230º has a negative x and y component x -y -x y 0º0º 270º 180º 90º

13. Example You walk northeast 4000 m. (That’s 45º north of east). How far east and how far north have you walked? N W S E 4000 m 45º

14. East: = 4000 cos 45 = 4000 x 0.707 = 2828 m North: = 4000 sin 45 = 4000 x 0.707 = 2828 m 2828m N 2828m E 4000 m 45º

16. Adding Vectors w/ Components 1. Find x and y components of all vectors 2. Add all x-components 3. Add all y-components 4. You now have r x and r y 5. Use Pythag. to find magnitude 6. Use θ = tan -1 (r y /r x ) to find the angle

17. Example 1: Add a and b a = 3.0 N @ 20º b = 2.0 N @ 45º a b +y +x

18. Example: Add a and b a = 3.0 N @ 20ºb = 2.0 N @ 45º a x = 3 cos 20ºb x = 2 cos 45º = 2.82 N (x)= 1.41 N (x) a y = 3 sin 20ºb y = 2 sin 45º = 1.03 N (y)= 1.41 N (y)

19. Add x and y components a x + b x =2.82 N + 1.41 N = 4.23 N (x)= r x a y + b y = 1.03 N + 1.41 N = 2.44 N (y)= r y

20. Use Pythag. to find resultant r r 2 = (r x ) 2 + (r y ) 2 r 2 = (4.23) 2 + (2.44) 2 r 2 = 23.85 r = 4.88 N r x = 4.23 r y = 2.44 r θ

21. Use θ = tan -1 (r y /r x ) to find angle θ = tan -1 (r y /r x ) = tan -1 (2.44/4.23) = tan -1 (0.577) θ = 30º r x = 4.23 r y = 2.44 r θ

22. Final Result a b +y +x r = 4.88 @ 30º b 30º

23. a = 4.0 m/s @ 35º b = 5.6 m/s @ 140º a x = 3.28 m/s b x = -4.29 m/s a y = 2.29 m/s b y = 3.60 m/s a x + b x = -1.01 m/s (x)= r x a y + b y = 5.89 m/s (y)= r y a b +y +x -x Example 2

24. r 2 = (-1.01) 2 + (5.89) 2 r 2 = 35.71 r = 5.98 N θ = tan -1 (r y /r x ) = tan -1 (5.89/-1.01) = tan -1 (-5.83) θ = -80.3º ??? θ = 180 + -80.3º = 99.7 r x = -1.01 r y = 5.89 r = 5.98 θ

25. Final Result a b +y +x a r θ θ = 99.7