1. Vectors You will be tested on your ability to: 1.correctly express a vector as a magnitude and a direction 2. break vectors into their components 3.add and multiply vectors 4.apply concepts of vectors to linear motion equations (ch. 2)

2. Vector vs. Scalar Scalar units are any measurement that can be expressed as only a magnitude (number and units) –Examples: 14 girls $85 65 mph Vector quantities are measurements that have BOTH a magnitude and direction. –Examples: Position Displacement Velocity Acceleration Force

3. Representing Vectors Graphically, vectors are represented by arrows, which have both a length (their magnitude) and a direction they point. v = 45 m/s v = 25 m/s d = 50 m a= 9.8 m/s 2

4. Representing Vectors The symbol for a vector is a bold letter. –For velocity vectors we write v –For handwritten work we use the letter with an arrow above it. v Algebraically –Vectors are written as a magnitude and direction –v = lvl, Θ –Example v = 25 m/s, 120 o or d = 50 m, 90 o

5. Drawing Vectors Choose a scale Measure the direction of the vector starting with east as 0 degrees. Draw an arrow to scale to represent the vector in the given direction Try it! v = 25 m/s, 190 o scale: 1 cm = 5 m/s This can be described 2 other ways –v = 25 m/s, 10 o south of west –v = 25 m/s, 80 o west of south Try d = 50 m, 290 o new scale?

6. Adding Vectors Vector Equation –v r = v 1 +v 2 –Resultant- the vector sum of two or more vector quantities. –Numbers cannot be added if the vectors are not along the same line because of direction! –Example…… –To add vector quantities that are not along the same line, you must use a different method…

7. An Example D1D1 D2D2 D3D3 DTDT D 1 = 169 km @ 90 degrees (North) D 2 = 171 km @ 40 degrees North of East D 3 = 195 km @ 0 degrees (East) D T = ???

8. Tip to tail graphical vector addition On a diagram draw one of the vectors to scale and label it. Next draw the second vector to scale, starting at the tip of the last vector as your new origin. Repeat for any additional vectors The arrow drawn from the tail of the first vector to the tip of the last represents the resultant vector Measure the resultant

9. Add the following d 1 = 30m, 60 o East of North d 2 = 20m, 190 o d r = d 1 +d 2 d r = ? d r = 13.1m, 61 o

10. Vector Subtraction Given a vector v, we define –v to be the same magnitude but in the opposite direction (180 degree difference) We can now define vector subtraction as a special case of vector addition. v 2 – v 1 = v 2 + (-v 1 ) Try this d 1 = 25m/s, 40 o West of North d 2 = 15m/s, 10 o 1cm = 5m/s Find : d r = d 1 +d 2 Find d r = d 1 - d 2 v –v–v

11. Multiplying a vector v by a scalar quantity c gives you a vector that is c times greater in the same direction or in the opposite direction if the scalar is negative. V cVcV -cV

12. Vector Components A vector quantity is represented by an arrow. v = 25 m/s, 60 o This single vector can also be represented by the sum of two other vectors called the components of the original. v = 50 m/s, 60 o sinΘ = V y / V V y = V sinΘ cosΘ = V x / V V x = V cosΘ

13. Try this: V 1 = 10 m @ 30 degrees above +x Find:V 1X = V 1Y = V 2 = 10 m @ 30 degrees above –x Find:V 2X = V 2Y = But V 2X should be NEGATIVE!!! Try using the angle 150 degrees for V 2 Ө1Ө1 Ө2Ө2

14. Try this: V 1 = 10 m @ 30 degrees above +x Find:V 1X = V 1Y = V 2 = 10 m @ 30 degrees above –x Find:V 2X = V 2Y = Find : V 3X = V 3Y = V 3 = 10 m @ 30 degrees below +x Try using the angle 330 degrees for V 3

15. Now try this: V X = 25m/s V Y = - 51m/s Find V=

16. and your point is??? ALWAYS: Describe a vector’s direction relative to the +x axis Measure counter-clockwise angles as positive Measure clockwise angles as negative

17. An Example D1D1 D2D2 D3D3 DTDT D 1 = 169 km @ 90 degrees (North) D 2 = 171 km @ 40 degrees North of East D 3 = 195 km @ 0 degrees (East) D T = ??? D 2X D 2Y D 1Y D 3X

18. A Review of an Example y (km) x (km)

19. But Wait... There’s more! y (km) x (km) We’ve Found: D TX = 326 km D TY = 279 km. For ID T I, use the Pythagorean Theorem. For the Direction of D T, use Tan -1

20. Practice it: Pg. 70, # 1, 4