## Calculus (3rd Edition)

$\begin{array}{*{20}{c}} {}&{Vector}&{Components}&{Length}\\ {\left( a \right)}&{4{\bf{i}} + 3{\bf{j}}}&{\left( {4,3} \right)}&5\\ {\left( b \right)}&{2{\bf{i}} - 3{\bf{j}}}&{\left( {2, - 3} \right)}&{\sqrt {13} }\\ {\left( c \right)}&{{\bf{i}} + {\bf{j}}}&{\left( {1,1} \right)}&{\sqrt 2 }\\ {\left( d \right)}&{{\bf{i}} - 3{\bf{j}}}&{\left( {1, - 3} \right)}&{\sqrt {10} } \end{array}$
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: $\begin{array}{*{20}{c}} {}&{Vector}&{Components}&{Length}\\ {\left( a \right)}&{4{\bf{i}} + 3{\bf{j}}}&{\left( {4,3} \right)}&5\\ {\left( b \right)}&{2{\bf{i}} - 3{\bf{j}}}&{\left( {2, - 3} \right)}&{\sqrt {13} }\\ {\left( c \right)}&{{\bf{i}} + {\bf{j}}}&{\left( {1,1} \right)}&{\sqrt 2 }\\ {\left( d \right)}&{{\bf{i}} - 3{\bf{j}}}&{\left( {1, - 3} \right)}&{\sqrt {10} } \end{array}$