Question

Find T, N, and B for the curve at the indicated point. [HINT: After finding T', plug in the specific value for t before computing N and B.] r(t) = t, t, et at (0, 0, 1)

Answer 1

Answer:

Step-by-step explanation:

Given that

$r(t) = (t,t,e^t)$

To find tangent, normal and binormal vectors at (0,0,1)

i) Tangent vector

$r'(t) = (1,1,e^t)$

At the particular point, r'(t) = (1,1,e)

Tangent vector = $\frac{r'(t)}{||r'(t)||} \\=\frac{(1,1,e)}{\sqrt{1+1+e^2} } \\=\frac{(1,1,e)}{\sqrt{2+e^2} }$

ii) Normal vector

T'(t) = $(0,0,e^t)$

At that point T'(t) = (0,0,e)/e = (0,0,1)

iii) Binormal

B(t) = TX N

= $\left[\begin{array}{ccc}i&j&k\\1&1&e^t\\0&0&e^t\end{array}\right] \\= e^t(i-j)$