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Question
Write the projection of \[\vec{b} + \vec{c} \text{ on } \vec{a} \text{ when } \vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \text{ and } \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .\]
Solution
\[\text{ Given that }\]
\[ \vec{a} = 2 \hat{i} - 2 \hat{j} + \hat{k} \]
\[ \vec{b} = \hat{i} + 2 \hat{j} - 2 \hat{k} \]
\[ \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} \]
\[ \vec{b} + \vec{c} = \hat{i} + 2 \hat{j} - 2 \hat{k} + 2 \hat{i} - \hat{j} + 4 \hat{k} = 3 \hat{i} + \hat{j} + 2 \hat{k} \]
\[\text{ Projection of } \vec{b} + \vec{c} \text{ on } \vec{a}\text{ is }\]
\[\frac{\left( \vec{b} + \vec{c} \right) . \vec{a}}{\left| \vec{a} \right|}\]
\[ = \frac{\left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) . \left( 2 \hat{i} - 2 \hat{j} + \hat{k} \right)}{2 \hat{i} - 2 \hat{j} + \hat{k}}\]
\[ = \frac{6 - 2 + 2}{\sqrt{4 + 4 + 1}}\]
\[ = \frac{6}{3}\]
\[ = 2\]
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