pt) Suppose r ? ( t ) = cos ( ? t ) i + sin ( ? t ) j + 2 t k represents the pos

Question

pt) Suppose r ? (t)=cos(?t)i+sin(?t)j+2tk represents the position of a particle on a helix, where z is the height of the particle.

(a) What is t when the particle has height 8 ?  
t=

(b) What is the velocity of the particle when  its height is 8 ?
v ? =

(c) When the particle has height 8 , it leaves the helix and moves along the tangent line at the constant velocity found in part (b).  Find a vector  parametric equation for the position of the particle in terms of the original parameter t as it moves  along this tangent line. In other words, the t value from part (a) should give you the same point on the line and the curve.

Explanation / Answer

(a) The variable z represents the height.
So, we need 4t = 8 ==> t = 2.

(b) v(t) = r'(t) = <-? sin(?t), ? cos(?t), 4>.
==> v(2) = <0, ?,2>.

(c) Since r(2) = <1, 0, 4>, and r'(2) = <0, -?,2>, the tangent line has equation
L(t) = <1, 0, 4> + t<0, -?, 2> = <1, -?t, 2t + 4>.

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