Question

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

Answer 1

We know that projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ = $\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$.

Let $\overrightarrow{a}=\hat{i}+3\hat{j}+7\hat{k}$ and $\overrightarrow{b}=2\hat{i}-3\hat{j}+6\hat{k}$.

∴ Projection of $\overrightarrow{a}$ on $\overrightarrow{b}$

$$\begin{gathered} =\frac{\left(\hat{i}+3\hat{j}+7\hat{k}\right).\left(2\hat{i}-3\hat{j}+6\hat{k}\right)}{\left|2\hat{i}-3\hat{j}+6\hat{k}\right|} \\ =\frac{1\times 2+3\times \left(-3\right)+7\times 6}{\sqrt{2^2+{\left(-3\right)}^2+6^2}} \\ =\frac{2-9+42}{\sqrt{49}} \\ =\frac{35}{7} \\ =5 \end{gathered}$$