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Vectors

# Vectors - Basis Vectors i = 1 0 0 j = 0 1 0 k = 0 0 1 u =...

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Unformatted text preview: Basis Vectors i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 u = u1 , u2 , u3 = u1 i + u2 j + u3 k Magnitude |u| = u2 + u2 + u 2 1 2 3 Curvature = dT 1 dT |v a| = = ds |v| dt |v|3 = |f (x)| [1 + (f (x))2 ]3/2 y = f (x) Principal Unit Normal Vector N= 1 dT dT/dt = ds |dT/dt| Osculating Circle radius: = 1 (t0 ) 1 N(t0 ) (t0 ) Dot Product u w = u1 w1 + u2 w2 + u3 w3 u w = |u||w| cos Projection projw u = Cross Product uw= i j k u1 u2 u3 w1 w2 w3 uw w ww center: C = r(t0 ) + Unit Binormal Vector B=TN Torsion =- dB N= ds x (t) y (t) z (t) x (t) y (t) z (t) x (t) y (t) z (t) |v a|2 |u w| = |u||w| sin Position, Velocity, Acceleration r(t) = x(t)i + y(t)j + z(t)k v(t) = r (t) = x (t)i + y (t)j + z (t)k a(t) = r (t) = x (t)i + y (t)j + z (t)k Arc Length b Acceleration a = aT T + aN N aT = va d |v| = dt |v| |a|2 - a2 = T |v a| |v| aN = |v|2 = Projectile Motion L= a b dx dt |v(t)| dt 2 + dy dt 2 + dz dt 2 dt r(t) = ((v0 cos )t + x0 ) i 1 + - gt2 + (v0 sin )t + y0 j 2 Gradient Vector L= a t s(t) = t0 |v( )| d, ds = |v(t)| dt f= f f f i+ j+ k x y z Unit Tangent Vector T= v dr = ds |v| Directional Derivative Du f = 1 ( f u) |u| ...
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