Equation of tangent to surface

Question

Equation of the tangent to the curve x=t,y=t^2,z=t^3 at its point M(1,1,1) is :(t=1) . I am confused with the fact that since this is a surface z=xy,therefore the equation should be of a plane but the answer given is a straight line in 3-d. I have also searched on google and where i found that they are taking it as vector r(t)= .

Answer 1

$\vec r(t)=(t,t^2,t^3)$

$\vec r'(t)=(1,2t,3t^2)$

$\vec u=\vec r'(1)=(1,2,3)$

the tangent line (red in the picture) at $A(1,1,1)$ is $A+s\vec u=(1+s,1+2s,1+3s)$

The curve is in blue

Answer 2

The equation you give, $[t,t^2,t^3]$ is the equation of a curve, not of a surface (because it has only one parameter, $t$. A surface would have two parameters). It has a tangent line, not a tangent plane.