Question

Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = 3t i + (7 − 4t) j + (3 + 2t) k

Answer 1

Answer:

The curve is given by r(s)=$3\frac{s}{\sqrt{29}} i + (7 - 4\frac{s}{\sqrt{29}} ) j + (3 + 2\frac{s}{\sqrt{29}} ) k$

Step-by-step explanation:

The aec length of the curve is given by:

L=$\int\limits^t_0 {|r'(t)|} \, dt$

r(t) = 3t i + (7 − 4t) j + (3 + 2t) k

r'(t)= 3 i -4 j + 2 k

|r'(t)|= $\sqrt{3^2+(-4)^2+2^2} =\sqrt{9+16+4} =\sqrt{29}$

L=$\int\limits^t_0 {\sqrt{29}} \, dt=\sqrt{29}\int\limits^t_0 \, dt$

L=$\sqrt{29}t|^t_0=\sqrt{29}t-\sqrt{29}0=\sqrt{29}t$

s(t)=$\sqrt{29}t$

We have to obtain t in terms of s, hence:

t=$\frac{s}{\sqrt{29}}$

Finally,

r(s)=$3\frac{s}{\sqrt{29}} i + (7 - 4\frac{s}{\sqrt{29}} ) j + (3 + 2\frac{s}{\sqrt{29}} ) k$