Unit Normal and Unit Binormal Vectors to a Space Curve

We have already looked at an important class of vectors known as unit tangent vectors denoted $\hat{T}(t)$. If $\vec{r}(t) = (x(t), y(t), z(t))$ is a vector-valued function for $t \in [a, b]$ that is differentiable, then the unit tangent vector at $t$ denoted $\hat{T}(t)$ represents the vector tangent to the point corresponding to $t$ and with length/magnitude $1$. We are now going to look at two other important vectors for space curves in $\mathbb{R}^3$.

Unit Normal Vectors

Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that represents the smooth curve $C$ for $t \in [a, b]$. Then for $\kappa (t) \neq 0$, the Unit Normal Vector (also known as the Unit Principle Normal Vector) denoted $\hat{N}(t) = \frac{\hat{T'}(t)}{\| \hat{T'}(t) \|}$ is a vector that is perpendicular to $\hat{T}(t)$ and has unit length/magnitude $1$.

We note that $\hat{N}(t) \perp \hat{T}(t)$ since $\| \hat{T}(t) \| = 1$, and so by one of the theorems on the Derivative Rules for Vector-Valued Functions page we have that $\hat{T}(t) \perp \hat{T'}(t)$ which implies that $\hat{T}(t) \perp \hat{N}(t)$. Geometrically, the unit normal vector at $t$ is a vector that is perpendicular to $\hat{T}(t)$ and points in the direction to which the curve $C$ is bending as illustrated in the following image.

Often times it can be extremely tedious to calculate unit normal vectors due to the frequent appearance of large numbers of terms and a radicals in the denominators that need differentiation.

Unit Binormal Vectors

We will now look at another important set of vectors known as unit binormal vectors.

Definition: Let $\vec{r}(t) = (x(t), y(t), z(t))$ be a vector-valued function that represents the smooth curve $C$ for $t \in [a, b]$. Then for $\kappa (t) \neq 0$ Unit Binormal Vector denoted $\hat{B}(t) = \hat{T}(t) \times \hat{N}(t)$, that is $\hat{B}(t)$ is the cross product of $\hat{T}(t)$ and $\hat{N}(t)$ and is perpendicular to both and has length/magnitude $1$.

The following diagram represents the unit binormal vectors for arbitrary points on a curve.