Question

Write the projection of the vector \[7 \hat{i} + \hat{j} - 4 \hat{k}\] on the vector \[2 \hat{i} + 6 \hat{j}+ 3 \hat{k} .\] 

Answer 1

\[\text{ Let }\vec{a} =7 \hat{i} + \hat{j} - 4 \hat{k} ; \vec{b} = 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \]
\[\text{ The projection of } \vec{a} \text{ on } \vec{b} \text{ is }\]
\[\left( \frac{\vec{a} . \vec{b}}{\left| \vec{b} \right|} \right)\]
\[ = \frac{\left( 7 \hat{i} + \hat{j} - 4 \hat{k} \right) . \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right)}{\left| 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right|}\]
\[ = \frac{14 + 6 - 12}{\sqrt{4 + 36 + 9}}\]
\[ = \frac{8}{7}\]