Question

Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P.

r(t)=ti+t^(2)j+tk, P(0,0,0)

Answer 1

T(t) = r'/|r'| = <1,2t,1>/√(4t^2+2)
P(0,0,0) = r(0)
T(0) = <1,0,1>/√2

r(0) = 0, so

the line is <0,0,0>+<1/√2,0,1/√2>t

Answer 2

How do you know that t=0 at (0,0,0)? How do you find it?

Answer 3

well, what value of t makes t and t^2 all zero?

Answer 4

r(t)=〈 2 sin t, 2 cos t, 4 sin^(2) t 〉 at the point (1, sqrt(3),1)

For this it is t=pi/6 to (1,sqrt(3),1). Can you show me how it is t=pi/6?

Answer 5

well, we want

2sint = 1, so sint = 1/2, so t=pi/6

Now it's just a matter of confirming that this also gives correct values for j and k.
It does.