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VECTORS in 3-D Space

VECTORS in 3-D Space. Vector Decomposition F x , F y , F z Addition of Vectors: Cartesian Vector Form: Unit Vectors:. Vectors in 3-D Space:. Vector Components F x = |F| cos  x F y = |F| cos  y F z = |F| cos  z. z. F. F z. y. F y. F x. x.

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VECTORS in 3-D Space

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  1. VECTORS in 3-D Space • Vector Decomposition • Fx, Fy, Fz • Addition of Vectors: • Cartesian Vector Form: • Unit Vectors:

  2. Vectors in 3-D Space: Vector Components Fx = |F| cos x Fy = |F| cos y Fz = |F| cos z z F Fz y Fy Fx x

  3. Cartesian Vector Form: Vector Components Fx = |F| cos x Fy = |F| cos y Fz = |F| cos z Magnitude Directional Cosines: Trigonometry Identity Cartesian Representation cos x = cos y = cos z= cos2x + cos2y + cos2z = 1

  4. Cartesian Representation • Cartesian Form:

  5. Different Problems • Case 1:

  6. Activity #1: • Write the vectors F1, F2 in cartesian vector: (just write equation without doing calculations). • Using Maple add vectorsF1, F2 to find R. • Using Maple Find Magnitude of R

  7. Unit Vector in Direction of F: • Cartesian Vector Form • Unit Vector, eF , in Direction of F: • Dividing Above Eqn by its magnitude: • But, since we know:

  8. Unit Vector • So, a unit vector is given by a vector F: • Or by its directional cosines: • To find

  9. Activity #2: • For previous activity find using MAPLE • (a) Unit vector in direction of F: eF • (b) Magnitude of the Unit Vector: eF • (c) Angle of Resultant force R:

  10. Unit Vector from Position • In some case the angles of a vector are not given, neither the components of force. • ONLY know: z A(3,-4,5 F=10N y x

  11. Activity #3 • Solve Problem 2.81 using MAPLE. • Follow the example given in Class • Class of Prob. 2.80

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