Question

Express the vector \[\vec{a} = 5 \text{i} - 2 \text{j} + 5 \text{k}\] as the sum of two vectors such that one is parallel to the vector \[\vec{b} = 3 \text{i} + \text{k}\]  and other is perpendicular to \[\vec{b}\]

Answer 1

\[\text{ Given that } \vec{a} =5 \text{i} - 2 \hat{j} + 5 \hat{k} \text{ and } \vec{b} =3\hat{i} + \hat{k} \]
\[\text{ Let } \vec{x} \text{ and } \vec{y} \text{ be such that } \]
\[ \vec{a} = \vec{x} + \vec{y} \]
\[ \Rightarrow \vec{y} = \vec{a} - \vec{x} . . . \left( 1 \right)\]
\[\text{ Since } \vec{x} \text{ is parallel to } \vec{b} ,\]
\[ \Rightarrow \vec{x} = t \vec{b}.................... \left( t \text{ is constant } \right)\]
\[ \Rightarrow \vec{x} = t \left( 3\hat{i} + \hat{k} \right) = 3t \hat{i} +t \hat{k} \]

\[\text{ Substituting the values of } \vec{x} \text{ and } \vec{a} \text{ in } (1), \text{ we get }\]
\[ \vec{y} = 5 \hat{i} - 2 \hat{j} + 5 \hat{k} - \left( 3t \hat{i} +t \hat{k} \right) = \left( 5 - 3t \right) \hat{i} - 2 \hat{j} + \left( 5 - t \right) \hat{k} . . . \left( 2 \right)\]
\[\text{ Since } \vec{y} \text{ is perpendicular to } \vec{b} ,\]
\[ \vec{y} . \vec{b} = 0\]
\[ \Rightarrow \left[ \left( 5 - 3t \right) \hat{i} - 2 \hat{j} + \left( 5 - t \right) \hat{k} \right] . \left( 3 \hat{i} + \hat{k} \right) = 0\]
\[ \Rightarrow 3 \left( 5 - 3t \right) + 0 + \left( 5 - t \right) = 0\]
\[ \Rightarrow 15 - 9t + 5 - t = 0\]
\[ \Rightarrow 20 - 10t = 0\]
\[ \Rightarrow t = 2\]
\[\text{ From } (1) \text{ and } (2), \text{ we get }\]
\[ \vec{x} = 6 \hat{i} +2 \hat{k} \]
\[ \vec{y} = - \hat{i} - 2 \hat{j} + 3 \hat{k}\]