First slide

Vectors

Question

With respect to a rectangular cartesian coordinate 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 and Z-axis respectively. The unit vector along the direction of sum of these vector is

Moderate

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

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

Solution

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

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