Show that the curvature of a line is $0$.

Solution For a line, the unit tangent vector $\mathbf{T}$ is along the constant direction of the line, so $\mathbf{T}$ is a constant and $\dfrac{d\mathbf{T}}{ds}=\mathbf{0}$. The curvature $\kappa\;=\;\left\Vert \dfrac{d\mathbf{T}}{ds}\right\Vert$ of a line is $0$.