Answer:
The curve is given by r(s)=[tex]3\frac{s}{\sqrt{29}} i + (7 - 4\frac{s}{\sqrt{29}} ) j + (3 + 2\frac{s}{\sqrt{29}} ) k[/tex]
Step-by-step explanation:
The aec length of the curve is given by:
L=[tex]\int\limits^t_0 {|r'(t)|} \, dt[/tex]
r(t) = 3t i + (7 − 4t) j + (3 + 2t) k
r'(t)= 3 i -4 j + 2 k
|r'(t)|= [tex]\sqrt{3^2+(-4)^2+2^2} =\sqrt{9+16+4} =\sqrt{29}[/tex]
L=[tex]\int\limits^t_0 {\sqrt{29}} \, dt=\sqrt{29}\int\limits^t_0 \, dt[/tex]
L=[tex]\sqrt{29}t|^t_0=\sqrt{29}t-\sqrt{29}0=\sqrt{29}t[/tex]
s(t)=[tex]\sqrt{29}t[/tex]
We have to obtain t in terms of s, hence:
t=[tex]\frac{s}{\sqrt{29}}[/tex]
Finally,
r(s)=[tex]3\frac{s}{\sqrt{29}} i + (7 - 4\frac{s}{\sqrt{29}} ) j + (3 + 2\frac{s}{\sqrt{29}} ) k[/tex]
