Question

If \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} ,\]  find the projection of \[\vec{a} \text{ on } \vec{b}\] 

Answer 1

\[\text{ We have }\]
\[ \vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = - \hat{j} + \hat{k} \]
\[\text{ The projection of } \vec{a} \text{ on } \vec{b} \text{ is }\]
\[\frac{\vec{a} . \vec{b}}{\left| \vec{b} \right|}\]
\[ = \frac{\left( \hat{i} - \hat{j} \right) . \left( - \hat{j} + \hat{k} \right)}{\left| - \hat{j} + \hat{k}90 \right|}\]
\[ = \frac{0 + 1 + 0}{\sqrt{1 + 1}}\]
\[ = \frac{1}{\sqrt{2}}\]