To find length of curve use:
[tex] \int\limits^a_b {|r'(t)|} \, dt [/tex]
where r(t) is vector equation:
[tex]r(t) = (1,-5,-2) + (-1,1,2)t[/tex]
[tex]r'(t) = (-1,1,2)[/tex]
Magnitude of vector r'(t) is then:
[tex] \sqrt{(-1)^2 +1^2 +2^2} =\sqrt{6} [/tex]
Now evaluate integral
[tex] \int\limits^3_1 {\sqrt{6}} \, dt = \sqrt{6} (3-1) = 2 \sqrt{6} [/tex]
