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Vector Product

Vector Product

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Vector Product

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  1. Vector Product Results in a vector

  2. Dot product (Scalar product) • Results in a scalar a · b = axbx+ayby+azbz Scalar

  3. Vector Product • Results in a vector =

  4. Properties… • a x b = - b x a • a x a = 0 • a x b = 0 if a and b are parallel. • a x (b + c) = a x b + a x c • a x(λb)= λ a x b

  5. Examples… • i x j = k, j x k = i, k x i = j • j x i = - k, i x k = -j, k x j = i • i x i = 0

  6. c a b θ Vector product c = a x b • c is perpendicular to a and b, in the direction according to the right-handed rule.

  7. c c a a b b Vector product – Direction: right-hand rule c = a x b θ

  8. c a b Vector product – right-hand rule c = a x b c a θ b

  9. Vector product – right-hand rule c = a x b c c b b θ a a

  10. Vector Product-magnitude a = (a1, 0, 0) b = (b1, b2, 0) c a x θ b2 b y

  11. Invariance of axb • The direction of axb is decided according the right-hand rule. • The magnitude of axb is decided by the magnitudes of a and b and the angle between a and b. axb is invariant with respect to changes from one right-handed set of axes to another.

  12. Application—Moment (torque) of a force F M O R

  13. Moment of a force about a point M = | F |d F M = | F | |R |sinθ O R M = RxF θ d

  14. Component of a vector a in an arbitrary direction s a s as --- Unit vector in the direction of s ax x

  15. Example--Component of a Force F in an arbitrary direction s F s Fs --- Unit vector in the direction of s

  16. Example--Component of a Moment M in an arbitrary direction s M s Fs --- Unit vector in the direction of s ---- Scalar Triple product

  17. Scalar Triple Product Scalar

  18. Volume of a parallelepiped = Volume of the parallelepiped. F E G bxc H a θ θ b D α C c A B

  19. Moment of a force about an axis A F s ---- Moment of F about axis AA’ --- Unit vector in the direction of s A’

  20. Vector Triple Product Vector