If $\vec{a}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=6\hat{i}-3\hat{j}+2\hat{k}$, then the projection of $\vec{a}$ on $\vec{b}$ is

Solution

1. The formula for the projection of vector $\vec{a}$ onto vector $\vec{b}$ is given by ${\text{proj}}_{\vec{b}}\vec{a}=\frac{\vec{a}\cdot \vec{b}}{|\vec{b}|}$.
2. Calculate the dot product: $$\begin{array}{rl}\vec{a}\cdot \vec{b}& =(2)(6)+(2)(-3)+(1)(2)\\ & =12-6+2\\ & =8\end{array}$$.
3. Calculate the magnitude of vector $\vec{b}$: $$\begin{array}{rl}|\vec{b}|& =\sqrt{6^2+(-3{)}^2+2^2}\\ & =\sqrt{36+9+4}\\ & =\sqrt{49}\\ & =7\end{array}$$.
4. Substitute these values into the formula to get $\frac{8}{7}$.
5. Hence the answer is (B).