Answer
The decomposition of ${\bf{a}}$ at $t=-1$:
${\bf{a}} = {a_{\bf{T}}}{\bf{T}} + {a_{\bf{N}}}{\bf{N}}$
$\left( { - 2,0} \right) = - \frac{2}{{\sqrt {10} }}{\bf{T}} + \frac{3}{5}\sqrt {10} {\bf{N}}$,
where ${\bf{T}} = \left( {\frac{1}{{\sqrt {10} }}, - \frac{3}{{\sqrt {10} }}} \right)$ and ${\bf{N}} = \left( { - \frac{3}{{\sqrt {10} }}, - \frac{1}{{\sqrt {10} }}} \right)$.
Work Step by Step
We have ${\bf{r}}\left( t \right) = \left( {\frac{1}{3}{t^3},1 - 3t} \right)$. The velocity and acceleration vectors are ${\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) = \left( {{t^2}, - 3} \right)$ and ${\bf{a}}\left( t \right) = {\bf{r}}{\rm{''}}\left( t \right) = \left( {2t,0} \right)$, respectively.
At $t=-1$, we get ${\bf{v}}\left( { - 1} \right) = \left( {1, - 3} \right)$ and ${\bf{a}}\left( { - 1} \right) = \left( { - 2,0} \right)$. Thus, the unit tangent vector is
${\bf{T}} = \frac{{\bf{v}}}{{||{\bf{v}}||}} = \frac{{\left( {1, - 3} \right)}}{{\sqrt {\left( {1, - 3} \right)\cdot\left( {1, - 3} \right)} }} = \left( {\frac{1}{{\sqrt {10} }}, - \frac{3}{{\sqrt {10} }}} \right)$
By Eq. (2) of Theorem 1 we have
${a_{\bf{T}}} = {\bf{a}}\cdot{\bf{T}} = \left( { - 2,0} \right)\cdot\left( {\frac{1}{{\sqrt {10} }}, - \frac{3}{{\sqrt {10} }}} \right)$
${a_{\bf{T}}} = - \frac{2}{{\sqrt {10} }}$
Next, we use Eq. (3) of Theorem 1 to find
${a_{\bf{N}}}{\bf{N}} = {\bf{a}} - {a_{\bf{T}}}{\bf{T}}$
${a_{\bf{N}}}{\bf{N}} = \left( { - 2,0} \right) - \left( { - \frac{2}{{\sqrt {10} }}} \right)\left( {\frac{1}{{\sqrt {10} }}, - \frac{3}{{\sqrt {10} }}} \right)$
${a_{\bf{N}}}{\bf{N}} = \left( { - 2,0} \right) + \left( {\frac{1}{5}, - \frac{3}{5}} \right) = \left( { - \frac{9}{5}, - \frac{3}{5}} \right)$
Since ${\bf{N}}$ is an unit vector, so
${a_{\bf{N}}} = ||{a_{\bf{N}}}{\bf{N}}|| = \sqrt {{{\left( { - \frac{9}{5}} \right)}^2} + {{\left( { - \frac{3}{5}} \right)}^2}} = \sqrt {\frac{{81}}{{25}} + \frac{9}{{25}}} $
${a_{\bf{N}}} = \frac{3}{5}\sqrt {10} $
To find ${\bf{N}}$ we use the equation
${\bf{N}} = \frac{{{a_{\bf{N}}}{\bf{N}}}}{{{a_{\bf{N}}}}} = \frac{5}{{3\sqrt {10} }}\left( { - \frac{9}{5}, - \frac{3}{5}} \right)$
${\bf{N}} = \left( { - \frac{3}{{\sqrt {10} }}, - \frac{1}{{\sqrt {10} }}} \right)$
Thus, we obtain the decomposition of ${\bf{a}}$ at $t=-1$:
${\bf{a}} = {a_{\bf{T}}}{\bf{T}} + {a_{\bf{N}}}{\bf{N}}$
$\left( { - 2,0} \right) = - \frac{2}{{\sqrt {10} }}{\bf{T}} + \frac{3}{5}\sqrt {10} {\bf{N}}$,
where ${\bf{T}} = \left( {\frac{1}{{\sqrt {10} }}, - \frac{3}{{\sqrt {10} }}} \right)$ and ${\bf{N}} = \left( { - \frac{3}{{\sqrt {10} }}, - \frac{1}{{\sqrt {10} }}} \right)$.
