An arrow shot from a bow travels along a curved trajectory given by the set of points satisfying

$\displaystyle P(x,y) = (0,22) + r(t)$

where $\displaystyle r(t) = <36\sqrt{2}t , (36t - 6t^2)>$

a. find the unit tangent vector T to this curve as a function of t

b. find unit nomrla vector N to this curve as function of t

c. taking {T,N} as an orthonormal basis defininf the orientation of the arrow, find the angle through which the arrow rotates from t=3 to t=4

d. using this angle, or otherwise, find a transformation matrix that maps the basis at time t=3 to the basis at time t=4

k so $\displaystyle r'(t) = <36\sqrt{2} , (36 - 12t) $

so a. $\displaystyle T = \frac{<36\sqrt{2} , (36 - 12t)>}{\sqrt{3888-864t+144t^2}} $

b. $\displaystyle N=\frac{<12t-36 , 36\sqrt{2}>}{\sqrt{3888-864t+144t^2}} $

c. i tried evaluating the values

$\displaystyle T(3) = \frac{<36\sqrt{2},0>}{\sqrt{2592}}$

$\displaystyle N(3)= \frac{<0,36\sqrt{2}>}{\sqrt{2592}}$

$\displaystyle T(4) = \frac{<36\sqrt{2},-12>}{\sqrt{2736}} $

$\displaystyle N(4) = \frac{<-12,36\sqrt{2}>}{\sqrt{2736}} $

but i don't know if this is usefull or where to go from there.

d. No idea.