Expert: Ahmed Salami - 11/24/2008

Question


A particle moves in the plane along a curve C with representation. r(t)=tcos(t)i+tsin(t)j for t≥0

(the i and j have a hat over them)

a. find the unit tangent and unit normal vectors T and N corresponding to t= pi/2 (both T and N have a hat over them.)



cos(t)-tsin(t) [i] + sin(t)+tcos(t) [j]
-------------------------------------------- =
sqrt [cos(t)-tsin(t)]^2 + [sin(t)+tcos(t)]^2

cos(t)-tsin(t) [i] + sin(t)+tcos(t) [j]
-------------------------------------------- =
sqrt { t^2 + 2t[cos(t)-sin(t)] }

cos(π/2)-(π/2)sin(π/2) [i] + sin(π/2)+(π/2)cos(π/2) [j]
----------------------------------------------------------=
sqrt { (π/2)^2 + 2(π/2)[cos(π/2)-sin(π/2)] }

       -(π/2) [i] + 1 [j]
T(π/2)= --------------------
        √[(π^2/4)-π]


okay so I have the first part I just dont understand on how to get the unit normal vectors! PLEASE HELP!


Answer
Hi Sam,
Very good work.
Lets consider the dot product of two vectors A and B.
A.B = |A||B|cos#
where |A| and |B| are the magnitudes of the vectors and # is the angle between them. If
A = ai + bj and
B = ci + dj
A.B = ac + bd
Note that the dot product (or scalar product) is not a vector but a scalar (a number). This method is used to find the angle between two vectors.
Now, the tangent and normal vectors at a point are perpendicular. So we know that the angle between them is π/2.
T.N = |T||N|cos(π/2)
T.N = 0

I'm sure you can complete it from here.
Regards.