Answer
The velocity vector:
${\bf{v}}\left( t \right) = \left( {{{\rm{e}}^t}, - 3,\frac{1}{2}{t^2} + t + \sqrt 2 } \right)$
Work Step by Step
Find the velocity vector:
${\bf{v}}\left( t \right) = \smallint {\bf{a}}\left( t \right){\rm{d}}t = \smallint \left( {{{\rm{e}}^t},0,t + 1} \right){\rm{d}}t = \left( {{{\rm{e}}^t},0,\frac{1}{2}{t^2} + t} \right) + {{\bf{c}}_0}$
The initial condition ${\bf{v}}\left( 0 \right) = \left( {1, - 3,\sqrt 2 } \right)$ gives
$\left( {1, - 3,\sqrt 2 } \right) = \left( {1,0,0} \right) + {{\bf{c}}_0}$
${{\bf{c}}_0} = \left( {0, - 3,\sqrt 2 } \right)$
Thus,
${\bf{v}}\left( t \right) = \left( {{{\rm{e}}^t},0,\frac{1}{2}{t^2} + t} \right) + \left( {0, - 3,\sqrt 2 } \right)$
${\bf{v}}\left( t \right) = \left( {{{\rm{e}}^t}, - 3,\frac{1}{2}{t^2} + t + \sqrt 2 } \right)$
