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. `-4hati-2hatj+5hatk`
B. `-4hati+2hatj+5hatk`
C. `4hati-2hatj+5hatk`
D. `-4hati-2hatj-5hatk`

Explanation

To solve the problem of finding the vector that must be added to the vectors \( \hat{i} - 3\hat{j} + 2\hat{k} \) and \( 3\hat{i} + 6\hat{j} - 7\hat{k} \) so that the resultant vector is a unit vector along the y-axis, we can follow these steps:

### Step 1: Define the Given Vectors
Let:
- Vector \( \mathbf{A} = \hat{i} - 3\hat{j} + 2\hat{k} \)
- Vector \( \mathbf{B} = 3\hat{i} + 6\hat{j} - 7\hat{k} \)

### Step 2: Find the Resultant Vector
The resultant vector \( \mathbf{R} \) when vectors \( \mathbf{A} \) and \( \mathbf{B} \) are added is given by:
\[ \mathbf{R} = \mathbf{A} + \mathbf{B} + \mathbf{r} \]
where \( \mathbf{r} \) is the vector we need to find.

### Step 3: Calculate \( \mathbf{A} + \mathbf{B} \)
Now, we will add vectors \( \mathbf{A} \) and \( \mathbf{B} \):
\[ \mathbf{A} + \mathbf{B} = (\hat{i} + 3\hat{i}) + (-3\hat{j} + 6\hat{j}) + (2\hat{k} - 7\hat{k}) \]
Calculating each component:
- For \( \hat{i} \): \( 1 + 3 = 4 \) so \( 4\hat{i} \)
- For \( \hat{j} \): \( -3 + 6 = 3 \) so \( 3\hat{j} \)
- For \( \hat{k} \): \( 2 - 7 = -5 \) so \( -5\hat{k} \)

Thus,
\[ \mathbf{A} + \mathbf{B} = 4\hat{i} + 3\hat{j} - 5\hat{k} \]

### Step 4: Set the Resultant Vector to be a Unit Vector along the y-axis
A unit vector along the y-axis is given by:
\[ \hat{j} \]
Therefore, we set up the equation:
\[ 4\hat{i} + 3\hat{j} - 5\hat{k} + \mathbf{r} = \hat{j} \]

### Step 5: Isolate \( \mathbf{r} \)
Rearranging the equation to solve for \( \mathbf{r} \):
\[ \mathbf{r} = \hat{j} - (4\hat{i} + 3\hat{j} - 5\hat{k}) \]
This simplifies to:
\[ \mathbf{r} = \hat{j} - 4\hat{i} - 3\hat{j} + 5\hat{k} \]
\[ \mathbf{r} = -4\hat{i} + (1 - 3)\hat{j} + 5\hat{k} \]
\[ \mathbf{r} = -4\hat{i} - 2\hat{j} + 5\hat{k} \]

### Final Result
The vector that must be added is:
\[ \mathbf{r} = -4\hat{i} - 2\hat{j} + 5\hat{k} \]