With respect to a rectangular Cartesian coordinate system, three vectors are expressed as
`vec(a)= 4hat(i)-hat(j), vec(b)= -3hat(i)+2hat(j) and vec(c )= -hat(k)`
Where, `hat(i), hat(j), hat(k)` are unit Vector, along the X, Y and Z-axis respectively. The unit vectors `hat(r )` along the direction of sum of these vector is

Options

A. `vec(r )= 1/(sqrt(3))(hat(i)+hat(j)-hat(k))`
B. `vec(r )= 1/(sqrt(2))(hat(i)+hat(j)-hat(k))`
C. `vec(r )= 1/3(hat(i)-hat(j)+hat(k))`
D. `vec(r )= 1/(sqrt(2))(hat(i)+hat(j)+hat(k))`

Explanation

To find the unit vector \(\hat{r}\) along the direction of the sum of the given vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we will follow these steps:

### Step 1: Write down the vectors
The vectors are given as:
\[ \vec{a} = 4\hat{i} - \hat{j} \]
\[ \vec{b} = -3\hat{i} + 2\hat{j} \]
\[ \vec{c} = -\hat{k} \]

### Step 2: Sum the vectors
We will sum the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):
\[ \vec{R} = \vec{a} + \vec{b} + \vec{c} \]
Substituting the values:
\[ \vec{R} = (4\hat{i} - \hat{j}) + (-3\hat{i} + 2\hat{j}) + (-\hat{k}) \]

### Step 3: Combine like terms
Now, we will combine the components along each axis:
- For \(\hat{i}\) components: \(4 - 3 = 1\)
- For \(\hat{j}\) components: \(-1 + 2 = 1\)
- For \(\hat{k}\) components: \(0 - 1 = -1\)

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

### Step 4: Write the resultant vector
The resultant vector can be expressed as:
\[ \vec{R} = \hat{i} + \hat{j} - \hat{k} \]

### Step 5: Find the magnitude of the resultant vector
The magnitude of \(\vec{R}\) is given by:
\[ |\vec{R}| = \sqrt{(1)^2 + (1)^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \]

### Step 6: Find the unit vector \(\hat{r}\)
The unit vector \(\hat{r}\) in the direction of \(\vec{R}\) is given by:
\[ \hat{r} = \frac{\vec{R}}{|\vec{R}|} = \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} \]

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

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