Answer
The decomposition of ${\bf{a}}$ at $t=2$:
${\bf{a}} = {a_{\bf{T}}}{\bf{T}} + {a_{\bf{N}}}{\bf{N}}$
$\left( {0,0,2} \right) = \frac{8}{{3\sqrt 2 }}{\bf{T}} + \frac{2}{3}{\bf{N}}$,
where ${\bf{T}} = \left( {\frac{{ - 1}}{{3\sqrt 2 }},\frac{1}{{3\sqrt 2 }},\frac{4}{{3\sqrt 2 }}} \right)$ and ${\bf{N}} = \left( {\frac{2}{3}, - \frac{2}{3},\frac{1}{3}} \right)$.
Work Step by Step
We have ${\bf{r}}\left( t \right) = \left( {4 - t,t + 1,{t^2}} \right)$. The velocity and acceleration vectors are ${\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) = \left( { - 1,1,2t} \right)$ and ${\bf{a}}\left( t \right) = {\bf{r}}{\rm{''}}\left( t \right) = \left( {0,0,2} \right)$, respectively.
At $t=2$, we get ${\bf{v}}\left( 2 \right) = \left( { - 1,1,4} \right)$ and ${\bf{a}}\left( 2 \right) = \left( {0,0,2} \right)$. Thus, the unit tangent vector is
${\bf{T}} = \frac{{\bf{v}}}{{||{\bf{v}}||}} = \frac{{\left( { - 1,1,4} \right)}}{{\sqrt {\left( { - 1,1,4} \right)\cdot\left( { - 1,1,4} \right)} }}$
${\bf{T}} = \left( {\frac{{ - 1}}{{3\sqrt 2 }},\frac{1}{{3\sqrt 2 }},\frac{4}{{3\sqrt 2 }}} \right)$
By Eq. (2) of Theorem 1 we have
${a_{\bf{T}}} = {\bf{a}}\cdot{\bf{T}} = \left( {0,0,2} \right)\cdot\left( {\frac{{ - 1}}{{3\sqrt 2 }},\frac{1}{{3\sqrt 2 }},\frac{4}{{3\sqrt 2 }}} \right)$
${a_{\bf{T}}} = \frac{8}{{3\sqrt 2 }}$
Next, we use Eq. (3) of Theorem 1 to find
${a_{\bf{N}}}{\bf{N}} = {\bf{a}} - {a_{\bf{T}}}{\bf{T}}$
${a_{\bf{N}}}{\bf{N}} = \left( {0,0,2} \right) - \frac{8}{{3\sqrt 2 }}\left( {\frac{{ - 1}}{{3\sqrt 2 }},\frac{1}{{3\sqrt 2 }},\frac{4}{{3\sqrt 2 }}} \right)$
${a_{\bf{N}}}{\bf{N}} = \left( {0,0,2} \right) - \left( { - \frac{4}{9},\frac{4}{9},\frac{{16}}{9}} \right) = \left( {\frac{4}{9}, - \frac{4}{9},\frac{2}{9}} \right)$
Since ${\bf{N}}$ is an unit vector, so
${a_{\bf{N}}} = ||{a_{\bf{N}}}{\bf{N}}|| = \sqrt {{{\left( {\frac{4}{9}} \right)}^2} + {{\left( { - \frac{4}{9}} \right)}^2} + {{\left( {\frac{2}{9}} \right)}^2}} $
${a_{\bf{N}}} = \frac{2}{3}$
To find ${\bf{N}}$ we use the equation
${\bf{N}} = \frac{{{a_{\bf{N}}}{\bf{N}}}}{{{a_{\bf{N}}}}} = \frac{3}{2}\left( {\frac{4}{9}, - \frac{4}{9},\frac{2}{9}} \right)$
${\bf{N}} = \left( {\frac{2}{3}, - \frac{2}{3},\frac{1}{3}} \right)$
Thus, we obtain the decomposition of ${\bf{a}}$ at $t=2$:
${\bf{a}} = {a_{\bf{T}}}{\bf{T}} + {a_{\bf{N}}}{\bf{N}}$
$\left( {0,0,2} \right) = \frac{8}{{3\sqrt 2 }}{\bf{T}} + \frac{2}{3}{\bf{N}}$,
where ${\bf{T}} = \left( {\frac{{ - 1}}{{3\sqrt 2 }},\frac{1}{{3\sqrt 2 }},\frac{4}{{3\sqrt 2 }}} \right)$ and ${\bf{N}} = \left( {\frac{2}{3}, - \frac{2}{3},\frac{1}{3}} \right)$.
