By Jazar R.N.
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Extra info for Advanced Dynamics
Commutativity: r1 + r2 = r2 + r1 . Associativity: r1 + (r2 + r3 ) = (r1 + r2 ) + r3 and c1 (c2 r) = (c1 c2 ) r. Distributivity: c (r1 + r2 ) = cr1 + cr2 and (c1 + c2 ) r = c1 r + c2 r. Identity element: r + 0 = r, 1r = r, and r − r = r + (−1) r = 0. Example 30 Nonorthogonal Coordinate Frame It is possible to deﬁne a coordinate frame in which the three scaled lines OA, OB, OC are nonorthogonal. 115) where, r1 , r2 , and r3 are constant. 120) is solvable provided bˆ1 · bˆ2 × bˆ3 = 0, which means bˆ1 , bˆ2 , bˆ3 are not coplanar.
When all of the vector characteristics—length, axis, end point, direction, and physical quantity—are speciﬁed, the vector is called a bounded vector, point vector, or vecpoint. Such a vector is ﬁxed at a point with no movability. 2. Vecline. If the start and end points of a vector are not ﬁxed on the vector axis, the vector is called a sliding vector, line vector, or vecline. A sliding vector is free to slide on its axis. 3. Vecface. When the affecting point of a vector can move on a surface while the vector displaces parallel to itself, the vector is called a surface vector or vecface.
The arc length between θ = 0 and θ = π/2 would then be one-fourth the perimeter of the circle. 251) where f (x, y, z) = 0 and g(x, y, z) = 0 represent two surfaces. The space curve would then be indicated by intersecting the surfaces. 252) where a and k are constant and ϕ is an angular variable. 260) Example 47 Two Points Indicate a Line Consider two points A and B with position vectors a and b in a coordinate frame. The condition for a point P with position vector r to lie on the line AB is that r − a and b − a be parallel.