Answer
$\begin{array}{*{20}{c}}
{}&{Vector}&{Components}&{Length}\\
{\left( a \right)}&{4{\bf{i}} + 3{\bf{j}}}&{\left( {4,3} \right)}&5\\
{\left( b \right)}&{2{\bf{i}} - 3{\bf{j}}}&{\left( {2, - 3} \right)}&{\sqrt {13} }\\
{\left( c \right)}&{{\bf{i}} + {\bf{j}}}&{\left( {1,1} \right)}&{\sqrt 2 }\\
{\left( d \right)}&{{\bf{i}} - 3{\bf{j}}}&{\left( {1, - 3} \right)}&{\sqrt {10} }
\end{array}$
Work Step by Step
We have ${\bf{i}} = \left( {1,0} \right)$ and ${\bf{j}} = \left( {0,1} \right)$.
(a) The components are
$4{\bf{i}} + 3{\bf{j}} = 4\left( {1,0} \right) + 3\left( {0,1} \right) = \left( {4,3} \right)$
The length: $||4{\bf{i}} + 3{\bf{j}}|| = \sqrt {16 + 9} = 5$.
(b) The components are
$2{\bf{i}} - 3{\bf{j}} = 2\left( {1,0} \right) - 3\left( {0,1} \right) = \left( {2, - 3} \right)$
The length: $||2{\bf{i}} - 3{\bf{j}}|| = \sqrt {4 + 9} = \sqrt {13} $.
(c) The components are
${\bf{i}} + {\bf{j}} = \left( {1,0} \right) + \left( {0,1} \right) = \left( {1,1} \right)$
The length: $||{\bf{i}} + {\bf{j}}|| = \sqrt {1 + 1} = \sqrt 2 $.
(d) The components are
${\bf{i}} - 3{\bf{j}} = \left( {1,0} \right) - 3\left( {0,1} \right) = \left( {1, - 3} \right)$
The length: $||{\bf{i}} - 3{\bf{j}}|| = \sqrt {1 + 9} = \sqrt {10} $.
In summary:
$\begin{array}{*{20}{c}}
{}&{Vector}&{Components}&{Length}\\
{\left( a \right)}&{4{\bf{i}} + 3{\bf{j}}}&{\left( {4,3} \right)}&5\\
{\left( b \right)}&{2{\bf{i}} - 3{\bf{j}}}&{\left( {2, - 3} \right)}&{\sqrt {13} }\\
{\left( c \right)}&{{\bf{i}} + {\bf{j}}}&{\left( {1,1} \right)}&{\sqrt 2 }\\
{\left( d \right)}&{{\bf{i}} - 3{\bf{j}}}&{\left( {1, - 3} \right)}&{\sqrt {10} }
\end{array}$
