Question

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

• 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. $\hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}+\hat{k})$

Answer 1

The correct option is A $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$

According to the given information:

The unit vector $\hat{r}$ along the direction of the sum of these vectors is given by-

The grad $|r|=\sqrt{1+1+1}=\sqrt{3}$

Now, the $\hat{r}=\frac{1}{\sqrt{3}}$

$\Rightarrow \frac{r}{grad|\hat{r}|}[ai-j-3i+2j-k]$

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