Find the vector that must be added to the vector `hat(i)-3hat(j)+2hat(k)` and `3hat(i)+6hat(j)-7hat(k)` so that the resultant vector is a unit vector along the y-axis.

Options

A. `4hat(i) + 2hat(j) + 5hat(k)`
B. `-4hat(i) - 2hat(j) + 5hat(k)`
C. `3hat(i) + 4hat(j) + 5hat(k)`
D. Null vector

Answer

To solve the problem, we need to find the vector \( \mathbf{r} \) that must be added to the two given vectors such that the resultant vector is a unit vector along the y-axis.

### Step-by-Step Solution:

1. **Identify the Given Vectors:**
The two vectors provided are:
\[ \mathbf{A} = \hat{i} - 3\hat{j} + 2\hat{k} \]
\[ \mathbf{B} = 3\hat{i} + 6\hat{j} - 7\hat{k} \]

2. **Add the Given Vectors:**
We need to find the resultant of these two vectors:
\[ \mathbf{A} + \mathbf{B} = (\hat{i} + 3\hat{i}) + (-3\hat{j} + 6\hat{j}) + (2\hat{k} - 7\hat{k}) \]
Simplifying this, we get:
\[ \mathbf{A} + \mathbf{B} = 4\hat{i} + 3\hat{j} - 5\hat{k} \]

3. **Set Up the Equation for the Resultant Vector:**
Let the vector we need to find be \( \mathbf{r} \). The equation for the resultant vector \( \mathbf{R} \) is:
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} + \mathbf{r} \]
We want \( \mathbf{R} \) to be a unit vector along the y-axis, which is represented as:
\[ \mathbf{R} = \hat{j} \]

4. **Substitute and Rearrange:**
Now substituting \( \mathbf{R} \) into the equation:
\[ \hat{j} = (4\hat{i} + 3\hat{j} - 5\hat{k}) + \mathbf{r} \]
Rearranging gives:
\[ \mathbf{r} = \hat{j} - (4\hat{i} + 3\hat{j} - 5\hat{k}) \]

5. **Simplify the Expression for \( \mathbf{r} \):**
Now, simplifying \( \mathbf{r} \):
\[ \mathbf{r} = \hat{j} - 4\hat{i} - 3\hat{j} + 5\hat{k} \]
This simplifies to:
\[ \mathbf{r} = -4\hat{i} + (1 - 3)\hat{j} + 5\hat{k} = -4\hat{i} - 2\hat{j} + 5\hat{k} \]

6. **Final Result:**
Therefore, the vector that must be added is:
\[ \mathbf{r} = -4\hat{i} - 2\hat{j} + 5\hat{k} \]