3D geometry: parametric equation and tangents

[AdityaDev]

I have a doubt in 3d geometry. I calculus and I know how to do partial derivatives.(but I don't know what it means).
If you have a parametric equation $x=t, y=t^2,z=t^3$ (the equation is randomn)
What does $\vec{r}=t\hat{i}+t^2\hat{j}+t^3\hat{k}$ represent?
now if it represents the position vector or the vector connecting origin and a point on the curve, then will $\frac{dr}{dt}$ give the tangent to the curve?

 

[Mark44]

I have a doubt in 3d geometry. I calculus and I know how to do partial derivatives.(but I don't know what it means).
If you have a parametric equation $x=t, y=t^2,z=t^3$ (the equation is randomn)
What does $\vec{r}=t\hat{i}+t^2\hat{j}+t^3\hat{k}$ represent?

It represents a curve in three-dimensional space. For each value of the parameter t, you get a vector from the origin to a point on the curve. To see what this curve looks like, plot 8 or 10 points and connect them.

now if it represents the position vector or the vector connecting origin and a point on the curve, then will $\frac{dr}{dt}$ give the tangent to the curve?

Yes