Question

With respect to a rectangular cartesian coordinate system, three vectors are expressed as $\overrightarrow{a}=4\hat{i}-\hat{j},\overrightarrow{b}=-3\hat{i}+2\hat{j}$ and
$\overrightarrow{c}=-\hat{k}$ where $\hat{i},\hat{j},\hat{k}$ are unit vectors, along the X, Y and Z-axis respectively. The unit vectors $\hat{r}$along the direction of sum of these vector 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. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$
• D. $\hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}+\hat{k})$
  

Answer 1

The correct option is A

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

$\overrightarrow{r}=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=4\hat{i}-\hat{j}-3\hat{i}+2\hat{j}-\hat{k}=\hat{i}+\hat{j}-\hat{k}$
$\hat{r}=\frac{\overrightarrow{r}}{|r|}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{1^2+1^2+(-1{)}^2}}=\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$.