Question

Find a tangent vector of unit length at the point with the given value of the parameter t. r(t) = 2 sin(t)i + 7 cos(t)j t = π/6

Answer 1

The tangent vector to r(t) at any t in the domain is

$\mathbf T(t)=\dfrac{\mathrm d\mathbf r(t)}{\mathrm dt}=2\cos t\,\mathbf i-7\sin t\,\mathbf j$

At t = π/6, the tanget vector is

$\mathbf T\left(\dfrac\pi6\right)=\sqrt3\,\mathbf i-\dfrac72\,\mathbf j$

To get the unit tangent, normalize this vector by dividing it by its magnitude:

$\left\|\mathbf T\left(\dfrac\pi6\right)\right\|=\sqrt{(\sqrt3)^2+\left(-\dfrac72\right)^2}=\dfrac{\sqrt{61}}2$

So the unit tangent at the given point is

$\dfrac{\mathbf T\left(\frac\pi6\right)}{\left\|\mathbf T\left(\frac\pi6\right)\right\|}=2\sqrt{\dfrac3{61}}\,\mathbf i-\dfrac7{\sqrt{61}}\,\mathbf j$