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Question
Write the projection of the vector \[\hat{i} + 3 \hat{j} + 7 \hat{k}\] on the vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\]
Solution
We know that projection of \[\vec{a}\] \[\vec{b}\]= \[\frac{\vec{a} . \vec{b}}{\left| \vec{b} \right|}\]
Let \[\vec{a} = \hat{i} + 3 \hat{j} + 7 \hat{k}\] and \[\vec{b} = 2 \hat{i} - 3 \hat{j} + 6 \hat{k}\]
∴ Projection of \[\vec{a}\] on \[\vec{b}\]
\[= \frac{\left( \hat{i} + 3 \hat{j} + 7 \hat{k} \right) . \left( 2 \hat{i} - 3 \hat{j} + 6 \hat{k} \right)}{\left| 2 \hat{i} - 3 \hat{j} + 6 \hat{k} \right|}\]
\[ = \frac{1 \times 2 + 3 \times \left( - 3 \right) + 7 \times 6}{\sqrt{2^2 + \left( - 3 \right)^2 + 6^2}}\]
\[ = \frac{2 - 9 + 42}{\sqrt{49}}\]
\[ = \frac{35}{7}\]
\[ = 5\]
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