Question

For the following parameterized​ curve, find the unit tangent vector. Bold r (t )r(t)equals=left angle 4 cosine (t )comma 4 sine (t )comma cosine (t )right angle4cos(t),4sin(t),cos(t)​, for 0 less than or equals t less than or equals pi

Answer 1

Answer:

T(t) $=\frac{\langle-4sin(t),4cos(t),-sin(t)\rangle}{\sqrt{16+sin^2t}}$

Step-by-step explanation:

To find the unit tangent vector for $r(t)=\langle4cos(t),4sin(t),cos(t)\rangle$

Unit Tangent Vector = $\frac{r^{I}(t) }{||r^{I}(t) ||}$

Where$r^{I}(t)$ is the derivative of r(t) and $||r^{I}(t)||$ is its modulus.

$r^{I}(t)=\langle-4sin(t),4cos(t),-sin(t)\rangle$

$||r^{I}(t)||=\sqrt{(-4sin(t))^2+(4cos(t))^2+(-sin(t))^2}$

$=\sqrt{16sin^2t+16cos^2t+sin^2t} \\=\sqrt{16(sin^2t+cos^2t)+sin^2t} \\Since sin^2t+cos^2t=1\\=\sqrt{16+sin^2t}$

Therefore, Unit Tangent Vector T(t) $=\frac{\langle-4sin(t),4cos(t),-sin(t)\rangle}{\sqrt{16+sin^2t}}$ for $0\leq t\leq \pi$