Write the projection of the vector $(\vec{b} + \vec{c})$ on the vector $\vec{a}$, where $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$.

2

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

Given vectors, $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, 

$\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$, 

$\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$. 

$\vec{b} + \vec{c} =(\hat{i} + 2\hat{j} - 2\hat{k})+(2\hat{i} - \hat{j} + 4\hat{k})$

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

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

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

$= \frac{6 - 2 + 2}{\sqrt{9}} = \frac{6}{3} = 2$