1. Vexing Vectors or Trig making its way into Physics

2. Upon Further Review By now you should the Pythag. Thm. –c 2 = a 2 + b 2 where c is the hypotenuse a, b are the sides You should have also learned the basic trig functions sine, cosine, tangent

3. In action sin  =  cos  =  tan  = OPP HYP ADJ HYP OPP ADJ  b a c y r r r y x y x x

4. Apply to Physics Instead of x,y,r we have –x,y,|  x| –v x,v y,|v|  |  x| x y |v| vxvx vyvy NOTE: v is a vector

5. Adding Vectors Add these pairs of these vectors: 4 m/sec3 m/sec 4 m/sec3 m/sec 4 m/sec 3 m/sec 4 m/sec 3 m/sec = = = =

6. Moving Right Along Vectors can be moved, as long as you do not change the direction or magnitude

7. Which means? 4 m/sec 3 m/sec = 4 m/sec 3 m/sec 4 m/sec 3 m/sec This new vector is called the RESULTANT. The resultant vector is the new vector that is formed when you add two (or more) vectors together.

8. Magnificent Magnitude We can find the magnitude of the resultant We can also find the direction 4 m/sec 3 m/sec 

9. More Results To find the resultant, use the Pythag. Thm! –c 2 = a 2 + b 2 To find the direction, find theta –use your inverse functions Calculators: –TI »2 nd, tan, ( v y / v x ), enter –Scientific »( v y / v x ), 2 nd, tan 4 m/sec 3 m/sec 

10. Resolve these issues What if you are given the magnitude (hypotenuse) and a direction? We can then find the “legs of the right triangle” –in physics, we call them the x and y components of the vector –this is called RESOLVING the vector ? ? 53 deg 25 m/sec

11. I’ve seen this before... Use the trig identities 53 deg 25 m/sec vxvx vyvy sin  = vyvy |v| vxvx cos  = |v|

12. This doesn’t add up (CP) To add vectors, first resolve the vectors –find the x,y compnents Add the x’s, add the y’s These give the x and y compneonts of the new vector “Hat Notation”

13. A very Graphic slide (CC) You can also add vectors graphically –Archaic method and a real pain! First place vectors head to tail –place the tail of one vector on the head of the other remember that vectors can be moved –draw a arrow (vector) from the tail of the first vector to the head of the second vector This is the resultant

14. Today’s practice (CC) Head to Tail 4 m/sec 3 m/sec

15. Sample Problems A preson walks 100 m to east and then turns left and walks an additional 200 m north. What is his displacement? If he wanted to go the shortest distance, what direction should he have walked in? A person runs 90 feet east, then 90 feet north. What is his displacement? What direction of he went in a straight line? A boat launches perpindicular to a river with a speed of 30 m/sec. If the river’s current is 10 m/sec downstream, then what is the boat’s resultant velocity? What direction does it go in? A arrow moves with a velocity of 30 m/sec at 30 degrees. What are the x,y components of the velocity? A bullet is fired from a mountain moves 350 m at an angle of –25 degrees. What are the x,y components of the bullet’s displacement?