Tangent, Normal, and Binormal Vectors

Given $\vec{r}(t)$, the derivative ${\vec{r}}^{\prime }(t)$ is the tangent vector. The tangent line to $\vec{r}(t)$ at $P$ is the line passing through P and parallel to $\vec{r}(t)$. Note that ${\vec{r}}^{\prime }(t)\ne 0$ (vectors must have magnitude)

Tangent vector:

$${\vec{r}}^{\prime }(t)$$

Unit tangent vector:

$$\vec{T}(t)=\frac{{\vec{r}}^{\prime }(t)}{\mid \mid {\vec{r}}^{\prime }(t)\mid \mid }$$

• Tangent to curve with magnitude 1

Unit normal vector:

$$\vec{N}(t)=\frac{{\vec{T}}^{\prime }(t)}{\mid \mid T(t)\mid \mid }$$

• Orthogonal to $\vec{T}$ (hence also orthogonal to original curve $\vec{r}(t)$)

Binormal vector:

$$\vec{B}(t)=\vec{T}(t)\times \vec{N}(t)$$

• Orthogonal to both the tangent and normal vector (definition of cross product basically)