`overset(rarr)A` is a vector which when added to the resultant of vectors `(2 hat i - 3 hat j + 4 hat k)` and `(hat i+5 hat j +2 hat k)` yields a unit vector along the y axis then vector `overset(rarr)A` is :

Options

A. `-3hati -hat j-6hatk`
B. `3 hat I +hat j-6 hat k`
C. `3 hat I - hat j +6 hatk`
D. `3 hat I +hat j +6 hatk`

Explanation

To solve the problem, we need to find the vector \(\overset{\rarr}{A}\) such that when it is added to the resultant of the two given vectors, it yields a unit vector along the y-axis.

Let's break down the solution step by step:

### Step 1: Find the resultant of the two given vectors
The two vectors given are:
1. \(\overset{\rarr}{V_1} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}\)
2. \(\overset{\rarr}{V_2} = \hat{i} + 5 \hat{j} + 2 \hat{k}\)

To find the resultant vector \(\overset{\rarr}{R}\), we add these two vectors:
\[ \overset{\rarr}{R} = \overset{\rarr}{V_1} + \overset{\rarr}{V_2} \]

Calculating the components:
- For \(\hat{i}\): \(2 + 1 = 3\)
- For \(\hat{j}\): \(-3 + 5 = 2\)
- For \(\hat{k}\): \(4 + 2 = 6\)

Thus, the resultant vector is:
\[ \overset{\rarr}{R} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \]

### Step 2: Set up the equation with vector \(\overset{\rarr}{A}\)
Let \(\overset{\rarr}{A} = a \hat{i} + b \hat{j} + c \hat{k}\).

According to the problem, when \(\overset{\rarr}{A}\) is added to \(\overset{\rarr}{R}\), the result should be a unit vector along the y-axis. A unit vector along the y-axis can be represented as:
\[ \overset{\rarr}{U} = 0 \hat{i} + 1 \hat{j} + 0 \hat{k} \]

So we have:
\[ \overset{\rarr}{R} + \overset{\rarr}{A} = \overset{\rarr}{U} \]

Substituting the expressions:
\[ (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) + (a \hat{i} + b \hat{j} + c \hat{k}) = 0 \hat{i} + 1 \hat{j} + 0 \hat{k} \]

### Step 3: Equate the components
Now we equate the components from both sides:
1. For \(\hat{i}\): \(3 + a = 0\)
2. For \(\hat{j}\): \(2 + b = 1\)
3. For \(\hat{k}\): \(6 + c = 0\)

### Step 4: Solve for \(a\), \(b\), and \(c\)
From the equations we get:
1. \(a = -3\)
2. \(b = 1 - 2 = -1\)
3. \(c = -6\)

### Step 5: Write the vector \(\overset{\rarr}{A}\)
Now substituting the values of \(a\), \(b\), and \(c\) back into the expression for \(\overset{\rarr}{A}\):
\[ \overset{\rarr}{A} = -3 \hat{i} - 1 \hat{j} - 6 \hat{k} \]

### Final Answer
Thus, the vector \(\overset{\rarr}{A}\) is:
\[ \overset{\rarr}{A} = -3 \hat{i} - 1 \hat{j} - 6 \hat{k} \]

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