# Find TNB Vectors for a given point

Can anyone tell me whether or not my work and answer below are correct? This is question 13.3.48 in Stewart Calculus 7th edition.

Here is the problem definition:
"Find the vectors $\vec T, \vec N, \vec B$ at the given point. $\vec r(t) = (\cos t)i +(\sin t)j + (ln \cos t)k$ , (1,0,0)"

Here is my work:
Note: $t=0 \text{ because } \cos 0 = 1, \sin 0 = 0,\text{ and }\ln[ \cos 0] = \ln(1)=0$

$\vec r'(t) = (-\sin t)i + (\cos t)j + (-\tan t)k$

$|\vec r'(t)| = \sec t$

$\vec T(t)= (-\sin t \cos t)i + ((\cos t)^2)j + (-\sin t)k$

$\vec T'(t) = (1-2(\cos t)^2)i -(2 \sin t \cos t)j -(\cos t)k$

$|\vec T'(t)| = \sqrt{1+(\cos t)^2}$

$\vec N(t) = {\vec T'(t)\over|\vec T'(t)|} = {(1-2(\cos t)^2)\over \sqrt{1+(\cos t)^2}}i + ({(-2(\sin t)(\cos t))\over \sqrt{1+(\cos t)^2}})j + {(-\cos t)\over \sqrt{1+(\cos t)^2}}k$

$\vec B(t) = \vec T(t) \times \vec N(t) = \begin{vmatrix} i & j & k \\ -\sin t \cos t & \cos^2(t) & -\sin t \\ {1-2(\cos t)^2\over \sqrt{1+(\cos t)^2}} & {-2(\sin t)(\cos t)\over \sqrt{1+(\cos t)^2}} & {-\cos t\over \sqrt{1+(\cos t)^2}} \end{vmatrix}$

Thus, for t=0 and point (1,0,0) we have:

$\vec T(0) = j$
$\vec N(0) = ({-\sqrt2\over 2})i + ({-\sqrt2\over 2})k$
$\vec B(0) = ({-\sqrt2\over 2})i + ({\sqrt2\over 2})k$

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You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". Here's a basic tutorial and quick reference. There's an "edit" link under the question. – joriki Sep 21 '12 at 22:08
Everything up to "abs(r'(t)) = sec t" seems OK (if by that you mean $|r'(t)|=\sec t$). But I don't see how you arrived at the expression for $T(t)$ in the next line. Shouldn't you just divide $r'(t)$ by $|r'(t)|$ do get $T(t)$? And $|T(t)|$ should be $1$, not $\sqrt{1+\cos^2t}$, no? – joriki Sep 21 '12 at 22:15
Strings like "sin" are interpreted as concatenations of variable names and are therefore italicized. To get the right font and spacing for function names like $\sin$, you can use the predefined commands like \sin, or if you need a function name like $\operatorname{erf}$ for which there's no predefined command, you can use \operatorname{erf}. To format square roots, use e.g. \sqrt{1+x^2} to get $\sqrt{1+x^2}$. – joriki Sep 21 '12 at 22:25
To include text like "omitting this because I don't know how to format it" in a formula, you can use \text{the text}. – joriki Sep 21 '12 at 22:41
@joriki, Thank you. I fixed the omission and cleaned up the formatting enough that it now accurately represents my work in a more easily readable manner. I may make more formatting changes later when I have time. Can you tell yet whether I got this correct? If not, where any math error might be? Thank you. – CodeMed Sep 21 '12 at 22:52