I am trying to find the unit tangent vector T(t)

Heres the problem

$\displaystyle r(t) = ti + \frac{1}{t}j, t = 1$

$\displaystyle v(t) = i - \frac {1}{t^2} j$

then

$\displaystyle a(t) = \frac {2}{t^3} j$

I think the magnitude is:

$\displaystyle ||v(t)|| = \sqrt \frac{2} {t^4} j$

then the solutions manual has this:

$\displaystyle T(t) = \frac {v(t)}{ || v(t) ||} = \frac{t^2} {\sqrt{ t^4 + 1}} (i - \frac{ 1}{t^2}j )$

I am not sure how they are getting this.

Can someone help me with the steps? I would appreciate this!!