Find the projection of vector $(\vec{b} + \vec{c})$ on vector $\vec{a}$, where $\vec{a} = 2\hat{i} + 2\hat{j} + \hat{k}, \vec{b} = \hat{i} + 3\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} + \hat{k}$.

4

The correct answer is Option (2) → 4 ##

$\vec{b} + \vec{c} = \hat{i} + 3\hat{j} + \hat{k} + \hat{i} + \hat{k} = 2\hat{i} + 3\hat{j} + 2\hat{k}$

Projection of $(\vec{b} + \vec{c})$ on $\vec{a} = \frac{(\vec{b} + \vec{c}) \cdot \vec{a}}{|\vec{a}|}$

$= \frac{(2\hat{i} + 3\hat{j} + 2\hat{k}) \cdot (2\hat{i} + 2\hat{j} + \hat{k})}{\sqrt{4 + 4 + 1}}$

$= \frac{4 + 6 + 2}{3} = \frac{12}{3} = 4$

Therefore, Projection of $(\vec{b} + \vec{c})$ on $\vec{a} = 4$