Tangent Vector

Tangent Vector

If $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ for $\alpha \le t\le \beta$ is the vector representation of a curve $C$, then at any point on $C$ at which $x^{\prime }(t),y^{\prime }(t),z^{\prime }(t)$ all exist and do not simultaneously vanish, we have:

$$\vec{T}={\vec{r}}^{\prime }$$

as a tangent vector to $C$.

Direction

$\vec{T}$ for a vector function $\vec{r}(t)$ always points in the direction in which the parameter $t$ increases along $C$.

Unit Tangent Vector

Tangent vector with length 1, denoted by $\hat{T}(t)$, such that:

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