Find the vectors T, N, and B at the given point.,
just need B
Vector T = 1/3<-2,2,1>,
Vector N = <-1/3,-2/3,2/3>
Here is a similar problem for reference
r(t) = t2, 2 / 3t3, t , (1, -2 / 3, -1) T = N = B = Find the vectors T, N, and B at the given point. r(t) = t2, 2 / 3 t3, t , (4, -16 / 3, -2) (4, -16 / 3, -2) corresponds to t = -2 T (t) = r'(t) / |r'(t)| = 2t, 2t2, 1 / root 4t2 + 4t4 + 1 = 2t, 2t2, 1 / 2t2 + 1, so T(-2) = -4 / 9, 8 / 9, 1 / 9. T'(t) = -4t(2t2 + 1)-2 2t, 2t2, 1 + (2t2 + 1)-1 2, 4t, 0 [by Formula 3 of the theoremdagger] = (2t2 + 1)-2 -8t2 + 4t2 + 2, -8t3 + 8t3 + 4t, -4t = 2(2t2 + 1)-2 1 – 2t2, 2t, -2t N(t) = T’ (t) / |T'(t)| = 2(2t2 + 1)-2 1 -2t2 , 2t, -2t / 2(2t2 + 1)-2 root(1 – 2t2)2 + (2t)2 + (-2t)2 = 1 – 2t2, 2t, -2t / root1 – 4t2 + 4t4 + 8t2 = 1 – 2t2, 2t, -2t / 1 + 2t2 N (-2) = -7 / 9, – 4 / 9, 4 / 9 and B (-2) = T(-2) times N (-2) = 4 / 9, 1 / 9, 8 / 9 .
Suppose u and v are differentiable vector functions, c is a scalar, and f is a real-valued function. Then d / dt [u(t) + v(t)] = u'(t) + v'(t) d / dt[cu(t)] = cu'(t) d / dt [f(t)u(t)] = f’ (t) u(t) + f(t) u’ (t) d / dt [u(t) Â· v(t)] = u'(t) Â· v(t) + u(t) Â· v'(t) d / dt [u(t) times v(t)] = u'(t) times v(t) + u(t) times v’ (t) d / dt [u(f(t))] = f'(t)u'(f(t))
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