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