# General vector equation proof

• Nov 21st 2009, 11:18 AM
Rubberduckzilla
General vector equation proof
A curve is defined by $r(t)$ such that $r'(t) \neq 0$
$T(t)=\frac{r'(t)}{|r'(t)|}$

Show that T and T' are always orthogonal by using |T|=1

edit: this should only take a few lines of working
i've tried using dot product =0 because the angle is 90 but i've no idea where to take that from there
• Nov 21st 2009, 12:51 PM
Opalg
Quote:

Originally Posted by Rubberduckzilla
A smooth curve is defined by $r(t)$ such that $r'(t) \neq 0$
The unit tangent vector is $T(t)=\frac{r'(t)}{|r'(t)|}$

Show that T and T' are always orthogonal by using |T|=1

edit: i've been told that this should only take a few lines of working
i've tried using dot product =0 because the angle is 90 but i've no idea where to take that from there

|T| = 1 tells you that T.T = 1. Differentiate that (product rule): T'.T + T.T' = 0. That's it (less than one line of working)!
• Nov 22nd 2009, 06:15 PM
lugncap
I accept with information:A curve is defined by r(t) such that r'(t) \neq 0
T(t)=\frac{r'(t)}{|r'(t)|}
Show that T and T' are always orthogonal by using |T|=1
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