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Vectors

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

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Vectors

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  1. Vectors You will be tested on your ability to: correctly express a vector as a magnitude and a direction break vectors into their components add and multiply vectors 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 d = 50 m v = 25 m/s a= 9.8 m/s2

  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, 120o or d = 50 m, 90o

  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, 190o scale: 1 cm = 5 m/s • This can be described 2 other ways • v = 25 m/s, 10o south of west • v = 25 m/s, 80o west of south • Try d = 50 m, 290o new scale?

  6. Adding Vectors • Vector Equation • vr = v1+v2 • 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 D3 D2 DT D1 D1 = 169 km @ 90 degrees (North) D3 = 195 km @ 0 degrees (East) D2 = 171 km @ 40 degrees North of East DT = ???

  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 • d1 = 30m, 60o East of North • d2 = 20m, 190o • dr = d1+d2 • dr = ? • dr = 13.1m, 61o

  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. • v2 – v1 = v2 + (-v1) • Try this • d1 = 25m/s, 40o West of North • d2 = 15m/s, 10o • 1cm = 5m/s • Find : dr = d1+d2 • Find dr = d1- d2 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. cV V -cV

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

  13. Try this: V2 = 10 m @ 30 degrees above –x Find: V2X = V2Y = V1 = 10 m @ 30 degrees above +x Find: V1X = V1Y = Ө2 Ө1 But V2X should be NEGATIVE!!! Try using the angle 150 degrees for V2

  14. Try this: V2 = 10 m @ 30 degrees above –x Find: V2X = V2Y = V1 = 10 m @ 30 degrees above +x Find: V1X = V1Y = Find : V3X = V3Y= V3 = 10 m @ 30 degrees below +x Try using the angle 330 degrees for V3

  15. Now try this: VX = 25m/s VY = - 51m/s Find V=

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

  17. An Example D3 D3X D2 D2Y D2X DT D1Y D1 D1 = 169 km @ 90 degrees (North) D3 = 195 km @ 0 degrees (East) D2 = 171 km @ 40 degrees North of East DT = ???

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

  19. But Wait. . . There’s more! We’ve Found: DTX = 326 km DTY = 279 km. y (km) For IDTI, use the Pythagorean Theorem. For the Direction of DT, use Tan-1 x (km)

  20. Practice it: • Pg. 70, # 1, 4

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