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Adding Vectors by Components

Adding Vectors by Components. Dimension. The number of coordinates needed to specify a point (Ex) (Ex) . 0. One dimension = position on a line Two dimensions = position on a plane Three dimensions = position in 3D-space Four dimensions = 3D + time. To describe a direction.

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Adding Vectors by Components

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  1. Adding Vectors by Components

  2. Dimension The number of coordinates needed to specify a point (Ex) (Ex) 0

  3. One dimension = position on a line • Two dimensions = position on a plane • Three dimensions = position in 3D-space • Four dimensions = 3D + time

  4. To describe a direction

  5. x degrees north of east = Start at east and rotate x degrees toward north. • x degrees east of north = Start at north and rotate x degrees toward east.

  6. Revisit Vectors 1) Vectors in one dimension Fnet = ? F1 = 120 N F2 = 80 N

  7. 2) Vectors in two dimensions Fnet = ?

  8. Component Vectors • Visualizing Vectors in 2 Dimensions | Mechanics | Khan Academy • Vector resolution • Breaking a vector into the x- and y- components • Rx = • Ry = • vector R = Rx = the x component of a vector, R Ry = the y component of a vector, R

  9. Example • Find the x- and y-component of R if R = 15 cm and ө = 50˚.

  10. Adding Vectors by Components Add the vectors. 15 m/s 13 m/s 20˚ 40˚

  11. y-axis 13 m/s resultant vector 15 m/s 20˚ 40˚ x-axis

  12. y-axis 20˚ 40˚ x-axis

  13. y-axis 20˚ 40˚ x-axis

  14. y-axis 13 m/s 13sin 20˚ 20˚ 13cos20˚ 15 m/s 15sin 40˚ 40˚ x-axis 15cos40˚

  15. y-axis 13sin 20˚ 20˚ 15sin 40˚ 40˚ x-axis 15cos40˚ 13cos20˚

  16. y-axis 13sin 20˚ resultant vector 15sin 40˚ x-axis 15cos40˚ 13cos20˚

  17. Example The table below represents a set of force vectors. These vectors begin at the origin of a coordinate system, and end at the coordinates given in the table. • What is the magnitude of the resultant of the sum of these three vectors? • What is the size of the angle, ө, that the resultant makes with the horizontal (x-axis)?

  18. A (5, 7) • Find the resultant vector. B(-8, -3) C(9, -10)

  19. Answer 8.5 R (6, -6)

  20. A • Find the resultant vector. 3 N 55° 40° 50° 4N 5N B C

  21. 25° 4.3N R (1.8, -3.9)

  22. Example A GPS receiver indicates that your home is 15.0 km and 40.0˚ north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? *north of west = north from west

  23. x km, B ө ˚ R, 15 km 5 km, A 40˚ Ax = 0 Bx Rx = 15 cos 140˚ = - 11.5 Ay = 5 By Ry = 15 sin 140˚ = 9.6 Ax + Bx = Rx 0 + Bx = - 11.5 Bx = - 11.5 Ay + By = Ry 5 + By = 9.6 By = 4.6

  24. x km, B ө ˚ 4.6 km - 11.5 km Ay + By = Ry 5 + By = 9.6 By = 4.6 Ax + Bx = Rx 0 + Bx = - 11.5 Bx = - 11.5 x = ө + 90˚ = cos-1 (-11.5/12.4) =158˚ ө = 158˚ - 90˚ = 68˚

  25. Example A football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled. What is the magnitude and direction of the runner’s total displacement?

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