2 questions on vector functions

[DWill]

Homework Statement

1) Find parametric equations for the line that is tangent to r(t) = (sin t)i + (t^2 - cos t)j + (e^t)k at the parameter value t = 0.

2) For the equation r(t) = (cos t)i + (sin t)j and for t >= 0, is the particle's acceleration vector always orthogonal to its velocity vector?

Homework Equations

The Attempt at a Solution

1) According to my text the tangent line to the curve r(t) = f(t)i + g(t)j + h(t)k is the line that passes through the point (f(t0), g(t0), h(t0)) parallel to v(t0), where t0 = 0 in this problem.
I solved the velocity equation v(t) = (cos t)i + (2t + sin t)j + (e^t)k, and v(0) = i + k.
This is the answer:
x = t, y = -1, z = 1+t
I found the point (f(0), g(0), t(0)) = (0, -1, 1), I'm not sure how to find the equation of a line that passes through this point and is parallel to another line? thanks

2) For this one I just want to make sure, to find where the acceleration vector is orthogonal to its velocity vector I find where their dot products equal 0, right?

 

[Dick]

The equation of a line passing through a point x(0) with instantaneous velocity v(0) (i.e. tangent line) is x(0)+v(0)*t. And yes, orthogonal means dot product zero.