With respect to a rectangular cartesian coordinate system, three vectors are expressed as `veca=4hati -hatj , vecb=-3hati" and "vecc=-hatk` where `hati,hatj,hatk` are unit vectors of axis x,y and z then `hatr` along the direction of sum of these vector is :-

Options

A. `hatr=(1)/(sqrt(3))(hati-hatj-hatk)`
B. `hatr=(1)/(sqrt(2))(hati+hatj-hatk)`
C. `hatr=(1)/(3)(hati-hatj+hatk)`
D. `hatr=(1)/(sqrt(2))(hati+hatj+hatk)`

Answer

To solve the problem, we need to find the resultant vector from the given vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), and then determine the unit vector in the direction of this resultant vector.

### Step 1: Write down the vectors
The vectors are given as:
\[ \vec{a} = 4\hat{i} - \hat{j} \]
\[ \vec{b} = -3\hat{i} \]
\[ \vec{c} = -\hat{k} \]

### Step 2: Sum the vectors
To find the resultant vector \(\vec{R}\), we add the three vectors together:
\[ \vec{R} = \vec{a} + \vec{b} + \vec{c} \]
Substituting the values:
\[ \vec{R} = (4\hat{i} - \hat{j}) + (-3\hat{i}) + (-\hat{k}) \]

### Step 3: Combine like terms
Now, we combine the components:
- For the \(\hat{i}\) component: \(4 - 3 = 1\)
- For the \(\hat{j}\) component: \(-1\)
- For the \(\hat{k}\) component: \(-1\)

Thus, we have:
\[ \vec{R} = 1\hat{i} - 1\hat{j} - 1\hat{k} \]
or simply:
\[ \vec{R} = \hat{i} - \hat{j} - \hat{k} \]

### Step 4: Find the magnitude of the resultant vector
The magnitude of \(\vec{R}\) is calculated as:
\[ |\vec{R}| = \sqrt{(1)^2 + (-1)^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 5: Find the unit vector in the direction of \(\vec{R}\)
The unit vector \(\hat{r}\) in the direction of \(\vec{R}\) is given by:
\[ \hat{r} = \frac{\vec{R}}{|\vec{R}|} \]
Substituting the values:
\[ \hat{r} = \frac{\hat{i} - \hat{j} - \hat{k}}{\sqrt{3}} \]

### Final Result
Thus, the unit vector along the direction of the sum of these vectors is:
\[ \hat{r} = \frac{1}{\sqrt{3}} \hat{i} - \frac{1}{\sqrt{3}} \hat{j} - \frac{1}{\sqrt{3}} \hat{k} \]