How to Find the Vector Equation of a Tangent Line to a Curve at a Given Point

[ElDavidas]

Can anybody help me out with this Q?

"A curve R in space has vector equation:

$$x = (sin(\pi u), u^2 - 1, u^2 + 3u + 3)$$

u is a real number. Find a vector equation of the tangent line to R at the point (0, 0, 1)"

 

[siddharth]

What are your ideas or thoughts on how to solve this problem? You need to show some work to get help.

 

[ElDavidas]

Well, I originally thought of taking the gradient of x and then plugging in the values of (0,0,1). Not sure if this is the right way to go about it though.

 

[0rthodontist]

The derivative of a curve at a point is indeed parallel to the tangent line (if the curve has nonzero speed at the point and a tangent line). But you can't just plug in (0, 0, 1) because you only have 1 variable-u. How do you find u such that x(u) = (0, 0, 1)?