2.7 Curves in three-dimensional space

We have seen the way to determine the tangent line to a curve in ${ℝ}^2$, and the different ways one can write the equation of the tangent line. We can either determine a vector $𝐯$ parallel to the line, and express the tangent line as $(x_0,y_0)+\lambda 𝐯$, $\lambda \in ℝ$; or we can determine a vector $𝐧$ normal to the line, and write the equation of the tangent line via the scalar product: $(x,y)\cdot 𝐧=(x_0,y_0)\cdot 𝐧$.

What happens for a (parametrized) curve in ${ℝ}^3$?

If $L$ is a straight line in ${ℝ}^3$, then we can see that there are infinitely many directions for normal vectors to $L$ in ${ℝ}^3$. Conversely, we can’t use a single normal vector to define a straight line as we could in ${ℝ}^2$. In fact, the set of points $(x,y,z)$ satisfying $(x,y,z)\cdot 𝐧=(x_0,y_0,z_0)\cdot 𝐧$ is not a line but a plane (as long as $𝐧$ is non-zero). On the other hand, we can use the derivative ${\gamma }^{\prime }(t)$ to write the equation of the tangent line to $\gamma$ at $\gamma (T)$:

Similarly, we can calculate the arc length of a parametrized curve between the points $\gamma (a)$ and $\gamma (b)$:

Thus the length of the arc from $\gamma (0)$ to $\gamma (\pi )$ is