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Vector Addition

Vector Addition. Cummutative Law . A. C. B. B. C. A. B + A = C. A + B = C. A + B = B + A. Vector Addition. Associative Law. C. C. (A + B) + C. A + (B + C). B + C. A + B. B. B. A. A. A. Vector Subtraction. Subtract B from A. A+B. B. A. -B. A-B. Right Hand Rule.

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Vector Addition

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  1. Vector Addition Cummutative Law A C B B C A B + A = C A + B = C A + B = B + A

  2. Vector Addition Associative Law C C (A + B) + C A + (B + C) B + C A + B B B A A A

  3. Vector Subtraction Subtract B from A A+B B A -B A-B

  4. Right Hand Rule E, B, F

  5. Right Hand Rule E, B, F

  6. Right Hand Rule Right Hand Rule for “F” Fingers in the direction of B. Thumb in the direction of I. Palm in the direction of F.

  7. Right Hand Rule Right hand rule for force: Fingers in the direction of B. Thumb in the direction of I. Palm in the direction of F.

  8. Right Hand Rule What happens when a current is driven through the wire? Force out of the screen. Force into the screen.

  9. Resolution of Vector Orthogonal Vectors ZY XZ ZYX A ^ z ^ ^ x y YX

  10. Resolution of Vector Component Vectors A = Ax + Ay + Az ^ ^ ^ A = xAx + yAy + zAz |A| =√Ax2 + Ay2 + Az2

  11. Vector Problems 1: ^ ^ ^ ^ • A = x6 + y3 B = x4 - y7 A + B = ? Magnitude of Resulting Vector ? Its angle with respect to x- axis ?

  12. Solution of Problem 1 Unit Vectors x  (6+4) =10 Unit Vectors y  (3-7) = -4 Magnitude of Resultant Vector C=√102 + (-4)2 = √116 = 10.8 Angle α =tan-1 (-4/10) = -21.8º

  13. Vector Problem 2: Addition of Three Vectors A = -x8 + y12; B = -x5 + y15; C = x7 – y9 Magnitude of Resultant Vector D ? Angle of D with x-axis ?

  14. Scalar Product Scalar / Dot Product of Two Vectors Product of their magnitudes multiplied by the cosine of the angle between the Vectors

  15. Orthogonal Vectors Angular Dependence

  16. Scalar Product Scalar Product of a Vector with itself ? A . A = |A||A| cos 0º = A2

  17. Scalar Product Scalar Product of a Vector and Unit vector ? x . A =|x||A|cosα = Ax Yields the component of a vector in a direction of the unit vector Where alpha is an angle between A and unit vector x ^ ^

  18. Scalar Product Scalar Product of Rectangular Coordinate Unit vectors? x.y = y.z = z.x = ? = 0 x.x = y.y = z.z = ? = 1

  19. Scalar Product Problem 3: A . B = ? ( hint: both vectors have components in three directions of unit vectors)

  20. Scalar Product Problem 4: A = y3 + z2; B= x5 + y8 A . B = ?

  21. Scalar Product Problem 5: A = -x7 + y12 +z3; B = x4 + y2 + z16 A.B = ?

  22. Cylindrical Coordinates r= = z = z

  23. Spherical Coordinates = = = x = r sin  cos  y = r sin  sin  z = r cos 

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