Question

With respect to a rectangular cartesian co-ordinate system three vectors are expressed as $\vec{a}=4 \hat{i}-\hat{j}$, $\vec{b}=-3 \hat{i}+2 \hat{j}$ and $\vec{c}=-\hat{k}$ where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors, along the $x, y, z$ axes respectively. The unit vector along the direction of the sum of these vectors is

• A. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$
• B. $\hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}-\hat{k})$
• C. $\hat{r}=\frac{1}{3}(\hat{i}-\hat{j}+\hat{k})$
• D. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

Answer 1

Answer: (A)

The sum of three vectors,
$\vec{r}=\vec{a}+\vec{b}+\vec{c}$
$=(4 \hat{i}-\hat{j})+(-3 \hat{i}+2 \hat{j})+(-\hat{k})=\hat{i}+\hat{j}-\hat{k}$
Unit vector, $\hat{r}=\frac{\vec{r}}{r}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{1^{2}+1^{2}+(-1)^{2}}}$
$=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$