Answer
(a) $4{\bf{i}} + 3{\bf{j}} - 2{\bf{k}} = \left( {4,3, - 2} \right)$
The length: $||4{\bf{i}} + 3{\bf{j}} - 2{\bf{k}}|| = \sqrt {29} $
(b) ${\bf{i}} + {\bf{j}} + {\bf{k}} = \left( {1,1,1} \right)$
The length: $||{\bf{i}} + {\bf{j}} + {\bf{k}}|| = \sqrt 3 $
(c) $4{\bf{j}} + 3{\bf{k}} = \left( {0,4,3} \right)$
The length: $||4{\bf{j}} + 3{\bf{k}}|| = 5$
(d) $12{\bf{i}} + 8{\bf{j}} - {\bf{k}} = \left( {12,8, - 1} \right)$
The length: $||12{\bf{i}} + 8{\bf{j}} - {\bf{k}}|| = \sqrt {209} $
Work Step by Step
We have ${\bf{i}} = \left( {1,0,0} \right)$, ${\bf{j}} = \left( {0,1,0} \right)$, ${\bf{k}} = \left( {0,0,1} \right)$.
(a) The components of $4{\bf{i}} + 3{\bf{j}} - 2{\bf{k}}$:
$4{\bf{i}} + 3{\bf{j}} - 2{\bf{k}} = 4\left( {1,0,0} \right) + 3\left( {0,1,0} \right) - 2\left( {0,0,1} \right) = \left( {4,3, - 2} \right)$
The length:
$||4{\bf{i}} + 3{\bf{j}} - 2{\bf{k}}|| = \sqrt {{4^2} + {3^2} + {{\left( { - 2} \right)}^2}} = \sqrt {29} $
(b) The components of ${\bf{i}} + {\bf{j}} + {\bf{k}}$:
${\bf{i}} + {\bf{j}} + {\bf{k}} = 1\left( {1,0,0} \right) + 1\left( {0,1,0} \right) + 1\left( {0,0,1} \right) = \left( {1,1,1} \right)$
The length:
$||{\bf{i}} + {\bf{j}} + {\bf{k}}|| = \sqrt {{1^2} + {1^2} + {1^2}} = \sqrt 3 $
(c) The components of $4{\bf{j}} + 3{\bf{k}}$:
$4{\bf{j}} + 3{\bf{k}} = 0\left( {1,0,0} \right) + 4\left( {0,1,0} \right) + 3\left( {0,0,1} \right) = \left( {0,4,3} \right)$
The length:
$||4{\bf{j}} + 3{\bf{k}}|| = \sqrt {{4^2} + {3^2}} = 5$
(d) The components of $12{\bf{i}} + 8{\bf{j}} - {\bf{k}}$:
$12{\bf{i}} + 8{\bf{j}} - {\bf{k}} = 12\left( {1,0,0} \right) + 8\left( {0,1,0} \right) - 1\left( {0,0,1} \right) = \left( {12,8, - 1} \right)$
The length:
$||12{\bf{i}} + 8{\bf{j}} - {\bf{k}}|| = \sqrt {{{12}^2} + {8^2} + {{\left( { - 1} \right)}^2}} = \sqrt {209} $
