Tangent Vectors

Observing the Tangent Vectors of a Vectors Valued Function

Below shows the graph of a vector valued function, but a vector values function is more than just a static graph.  If you hit the Animate button, you will see the tangent vector move and change as time, $t$ progresses.  You can also choose to observe the unit tangent vector.  

$\vecs{r}(t)=<t, \frac{1}{3}t^2>, -3<t<3$

$\vecs{r}(t)=<\frac{5\cos(t)}{t}, \frac{5\sin(t)}{t}>, 1<t<30$

$\vecs{r}(t)=<3\cos(5t)\cos(t), 3\cos(5t)\sin(t)>, 0<t<4$

$\vecs{r}(t)=<2\sin(t), 4\sin^2(t)>, 0<t<10$

$\vecs{r}(t)=<t^2, e^{t/2}>, -3<t<3$

$\vecs{r}(t)=<3\cos(-t), 4\sin(-t)>, 0<t<10$

$\vecs{r}(t)=<t\cos(t), t\sin(t)>, 0<t<7$

$\vecs{r}(t)=<t^2, \frac{1}{3}t^3>, -2<t<2$

$\vecs{r}(t)=<e^t, e^{-t}>, -2<t<2$

$\vecs{r}(t)=<\frac{10}{1+t^2}, \frac{10}{1+t^4}>, -1<t<3$

     Tangent Vector                Unit Tangent Vector

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