Question

With respect to a rectangular cartesian coordinate system, three vectors are expressed as

$\overrightarrow{a} = 4\widehat{i} - \widehat{j}$, $\overrightarrow{b} = - 3\widehat{i} + 2\widehat{j}$ and $\overrightarrow{c} = - \widehat{k}$

Where $\widehat{i},\widehat{j},\widehat{k}$are unit vectors, along the X, Y and Z-axis respectively. The unit vectors $\widehat{r}$along the direction of sum of these vector is

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

Answer 1

Answer: (A)

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

Answer 2

$\overrightarrow{r} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 4\widehat{i} - \widehat{j} - 3\widehat{i} + 2\widehat{j} - \widehat{k} = \widehat{i} + \widehat{j} - \widehat{k}$

$\widehat{r} = \frac{\overrightarrow{r}}{|r|} = \frac{\widehat{i} + \widehat{j} - \widehat{k}}{\sqrt{1^{2} + 1^{2} + ( - 1)^{2}}} = \frac{\widehat{i} + \widehat{j} - \widehat{k}}{\sqrt{3}}$