Expert: Scotto - 11/13/2008

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= pi/6   (pi meaning the symbol)
B.Find parametric equations of the tangent line to the curve at P.

sorry one more thing >= means greater than or equal!

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

Answer
Notes
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The hats aren't necessary if you change 'representation r(t)' to 'representation of the vector r(t)', but I probably still would have gotten it since that's what i, j, and k are commonly used for.
That would have let me know that r(t) was a vector as well.
By the way, ≥ is alt-242 and ≤ is alt-243, but >= and <= are OK.
Also, π is alt-227.  alt-# means hold down the alt while typing that number.  It's really easy using the numbers on the right side of the keyboard that are in a nice square box.  So lets answer your questions.

Answers
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A. Since we are given the position function, the velocity is the derivative of the position function, and the acceleration is the second derivative.  To find them both at π/6, put π/6 into each of the equations.

To find the magnitude, add up take the square of each of the i, j, and k componets and then take the square root.  For example, if you had r(t) = ai + bj + ck, the magnitude is √(a²+b²+c²).

B. Evaluate the fuction at π/6 to get the starting point.  Evaluate the derivative at π/6 to get the values for the slope of the line.
If the function is f(t) = a(t)i + b(t)j + c(t)k, then the equation of the tangent line at t=t0 would be
[a(t0)+t*a'(t0)]i + [b(t0)+t*b'(t0)]j + [c(t0)+t*c'(t0)]k .