1.)

r(t) = < t, t^2, 1/t >

r'(t) = < 1, 2t, -1/t^2>

r"(t) = < 0, 2, 1/t^4>

|r'(t)| = squartroot (1 + 4t^2 + 1/t^4)

(a) Find the tangential component of the particle’s acceleration vector at time t = 1.

r'(1) = < 1, 2, -1>

r"(1) = < 0, 2, 1>

|r'(1)|= squareroot ( 1 + 4 +1)

Tangential component = r'(1) dot r"(1) / |r'(1)| = 0 + 4 -1 = 3/ sqroot 6

(b) Find all values of t at which the particle’s velocity vector is orthogonal to the particle’s acceleration vector.

Is part a right?

How do I find part b?

2.)

<4t, 5t^3, 2t^2>

a.) Find the unit tangent vector T(t) at the point where t = 1.

r'(1) = <4, 15, 4>

r"(1) = <0, 30, 4>

T(1) = 32/(sqrt(916))

b.) Find the parametric equations of the tangent line the curve at the point where t = 1

Thanks in advanced