Show that any straight line has curvature zero. Conversely, if a smooth curve has curvature zero, then it must be a straight line. Hint: For the first part, recall that any straight line has a position vector F(t) = (a + bt)i + (d + ct)j + (h + kt)k. For the converse, if κ = 0, then T′(t) = 0.

Q: 2. C. b. Let (t)= and (u) = At what point P do these two space curves intersect. a. Find the tangent…

A: As per rule I need to solve only first three subparts.

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A: Question (1) (a) We need to find the unit tangent and unit normal vector T(t) and N(t) of the vector…

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Q: 이 x 2 + .dx 2. Jordr V1-p3

A: I have explained everything with in the solution .please go through that.

Q: Consider the vector function r(t) that has unit tangent vector 1 T(t)= (1,1,2t), 0≤t≤ 2. √1+5t²…

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Q: 86. The unit nomal vecter to the curwe x?+y2 + Z² = 4 at (1,1,1) is %3D fa) li+aj+k)NI fb) lit2j+k)…

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Q: Prove that if T+¬AFB and I+¬A¬B, then I EA

A: Given proposition is Γ+¬A-B and Γ+¬A-¬B To prove Γ-A Now,  1. Γ+¬A-B∧Γ+¬A-¬B…

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Q: (b) Show that cb (q-p).u≤|li|| dt. a

A: From the previous result (a), If u is a unit vector then,…

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A: Disclaimer: Since, you have requested for the answer of part 3, only part 3 has been answered.…

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Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

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Q: Prove that for the curve x = 31, y = 3 f², z = 2 8³, 3 (1+21²) 2 p=-o=

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Q: Let C be the curve connecting (0,0) to (1,2) to (0,6) to (0,0) using straight lines. (5x*y – 3x²y+…

A: Greens theorem based problem......