Is There a Simpler Way to Find a Normal Using the Given T-Value?

[kuahji]

Write a in the form a=a$$_{Y}$$T+a$$_{N}$$N without finding T and N at the given t-value.

r=t$$\widehat{i}$$+3t$$^{2}$$$$\widehat{j}$$+t$$^{2}$$/2$$\widehat{k}$$ , t=2

The first thing I did was to find the first derivative. Which turned out to be i+6tj+tk
Then I found the length $$\sqrt{1+37t^{2}}$$
Then I took the derivative of that to find a$$_{T}$$ 37t/$$\sqrt{37t^{2}+1}$$

So my question really is, for a$$_{N}$$ do I really have to plug that mess into $$\sqrt{|a|^{2}-a_{T}^{2}}$$

 

[lippyka]

or is there a shorter way?a=37t/\sqrt{37t^{2}+1}(i+6tj+tk)+a_{N}(-6tj+k/\sqrt{37t^{2}+1}) , t=2a_{N}=\sqrt{17/3+78/3}