Question

Reparametrized (i) the straight line α(t)= x+t v where x, v ∈R 3 by arclength, (ii) the helix r(t)=(cost,sint,t) by arclength.

Answer 1

To reparametrize the straight line α(t)= x+tv by arc length, calculate the derivatives of x, y, and z, and substitute them into the arc length formula. For the helix r(t)=(cost,sint,t), calculate the derivatives of x, y, and z, substitute them into the arc length formula, and integrate.

Step-by-step explanation:

To reparametrize the straight line α(t) = x + tv, we need to parameterize it by arc length. The arc length s is given by the formula: s = ∫ √(x'(t)^2 + y'(t)^2 + z'(t)^2) dt. So, let's calculate the derivatives of x, y, and z, and substitute them into the formula.

For the helix r(t) = (cos(t), sin(t), t), we can also reparametrize it by arc length. The arc length s is given by the formula: s = ∫ √(x'(t)^2 + y'(t)^2 + z'(t)^2) dt. Calculate the derivatives of x, y, and z, substitute them into the formula, and integrate.