Expert: Alon Mandes - 11/15/2008

Question
A particle moves in space along a curve with representation r(t)=t(i)+2sin(t)j+3cos(t)k for t≥0
* sorry but all i,j,k have the hats on top, but I cant do that on here.  And r at the very beginning has the little arrow.

A. find the velocity and acceleration at the point P corresponding to t= π/6   
B.Find parametric equations of the tangent line to the curve at P.



This is the last part of my homework, and I cannot figure this question out! Please help me!


Answer
Our 3D trajectory is r(t)=t [i] + 2sin(t) [j] + 3cos(t) [k].

A. To find velocity we need to derive r(r) at the point t= π/6.
r'(t)=1 [i] + 2cos(t) [j] - 3sin(t) [k],
r'(π/6)=1 [i] + 2cos(π/6) [j] - 3sin(π/6) [k],
r'(π/6)=1 [i] +  √2 [j] - 3 [k].

B. The tangent line is from the form of :
x=xo+at
y=yo+bt
z=zo+ct
Where [xo,yo,zo] is r(π/6), & (a,b,c) is (dx/dt,dy/dt,dz/dt) at the
point t= π/6. Let's calculate :
r(π/6)=π/6 [i] + 1 [j] + 3/√2 [k]. So,
xo=π/6
yo=1
zo=3/√2
a=x'(t)=1
b=y'(t)=2cos(t) -> b=2cos(π/6)=1/√2
c=z'(t)=-3sin(t) -> c=-3sin(π/6)=-3/2
Hence, the parametric equation of the tangent line will be :
(π/6+t) [i] + (1+t/√2) [j] + 3/√2-3t/2 [k]

Alon.