What vector must be added to the two vectors `2hati-hatj+3hatk` and `3hati-2hatj-2hatk`, so that the resultant may be a unit vector along `z`- axis

Options

A. `5hati +hatk`
B. `-5hati +3hatj`
C. `3hati +5hatk`
D. `-3hati +2hatk`

Explanation

To solve the problem, we need to find a vector \( \vec{C} \) that, when added to the two given vectors \( \vec{A} \) and \( \vec{B} \), results in a unit vector along the z-axis.

### Step-by-Step Solution:

1. **Identify the Given Vectors:**
- Let \( \vec{A} = 2\hat{i} - \hat{j} + 3\hat{k} \)
- Let \( \vec{B} = 3\hat{i} - 2\hat{j} - 2\hat{k} \)

2. **Add the Two Vectors:**
- We first find the resultant vector \( \vec{R} = \vec{A} + \vec{B} \).
- Performing the addition:
\[ \vec{R} = (2\hat{i} + 3\hat{i}) + (-\hat{j} - 2\hat{j}) + (3\hat{k} - 2\hat{k}) \]
\[ \vec{R} = (2 + 3)\hat{i} + (-1 - 2)\hat{j} + (3 - 2)\hat{k} \]
\[ \vec{R} = 5\hat{i} - 3\hat{j} + 1\hat{k} \]

3. **Set Up the Equation for the Resultant Vector:**
- We want the resultant vector \( \vec{R} + \vec{C} \) to equal a unit vector along the z-axis, which can be represented as \( \hat{k} \).
- Therefore, we have:
\[ \vec{R} + \vec{C} = \hat{k} \]

4. **Substituting the Resultant Vector:**
- Substitute \( \vec{R} \) into the equation:
\[ (5\hat{i} - 3\hat{j} + 1\hat{k}) + \vec{C} = \hat{k} \]

5. **Isolate \( \vec{C} \):**
- Rearranging gives:
\[ \vec{C} = \hat{k} - (5\hat{i} - 3\hat{j} + 1\hat{k}) \]
\[ \vec{C} = \hat{k} - 5\hat{i} + 3\hat{j} - 1\hat{k} \]
\[ \vec{C} = -5\hat{i} + 3\hat{j} + (1 - 1)\hat{k} \]
\[ \vec{C} = -5\hat{i} + 3\hat{j} \]

6. **Final Result:**
- The vector \( \vec{C} \) that must be added is:
\[ \vec{C} = -5\hat{i} + 3\hat{j} \]