Consider the helix r(t)=(cos(1t), sin(1t), 1t).Compute, at t= π/6:?
Consider the helix r(t)=(cos(1t), sin(1t), 1t).Compute, at t= π/6:
A. The unit tangent vector T=( ______ , _______ , _______ )
B. The unit normal vector N=( ______ , _______ , _______ )
C. The unit binormal vector B=(______ , ______ , _______ )
D. The curvature K= _________________
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A) r'(t) = <-sin t, cos t, 1>
==> ||r'(t)|| = √2.
So, T(t) = r'(t)/||r'(t)|| = (1/√2) <-sin t, cos t, 1>.
==> T(π/6) = (1/√2) <-1/2, √3/2, 1> = <-1/(2√2), √3/(2√2), 1/√2>.
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B) T'(t) = (1/√2) <-cos t, -sin t, 0>
==> ||T'(t)|| = 1/√2.
So, N(t) = T'(t)/||T'(t)|| = <-cos t, -sin t, 0>.
==> N(π/6) = <-√3/2, -1/2, 0>.
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C) B(π/6) = T(π/6) x N(π/6) = <1/(2√2), -√3/(2√2), 1/√2>.
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D) k(π/6) = ||T'(π/6)||/||r'(π/6)|| = (1/√2)/√2 = 1/2.
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I hope this helps!