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Question
Find λ when the projection of \[\vec{a} = \lambda \hat{i} + \hat{j} + 4 \hat{k} \text{ on } \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k}\] is 4 units.
Solution
\[\text{ We have }\]
\[a = \lambda \hat{i} + \hat{j}+ 4 \hat{k} \text{ and } b = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \]
\[\text{ The projection of } \vec{a} \text{ on } \vec{b} \text{ is }\frac{\vec{a} . \vec{b}}{\left| \vec{b} \right|}\]
\[\text{ Given that }\]
\[\frac{\vec{a} . \vec{b}}{\left| \vec{b} \right|} = 4\]
\[ \Rightarrow \frac{\left( \lambda \hat{i} + \hat{j} + 4 \hat{k} \right) . \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right)}{\left| 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right|}\]
\[ \Rightarrow \frac{2\lambda + 6 + 12}{\sqrt{4 + 36 + 9}} = 4\]
\[ \Rightarrow \frac{2\lambda + 18}{7} = 4\]
\[ \Rightarrow 2\lambda + 18 = 28\]
\[ \Rightarrow 2\lambda = 10\]
\[ \therefore \lambda = 5\]
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