# Thread: General vector equation proof

1. ## General vector equation proof

A curve is defined by $\displaystyle r(t)$ such that $\displaystyle r'(t) \neq 0$
$\displaystyle T(t)=\frac{r'(t)}{|r'(t)|}$

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

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

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)!

3. 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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