`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 \( \overset{\rarr}{A} \) that, when added to the resultant of the two given vectors, results in a unit vector along the y-axis. Let's break this down 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 component-wise:

\[ \overset{\rarr}{R} = \overset{\rarr}{V_1} + \overset{\rarr}{V_2} \]

Calculating each component:
- The x-component: \( 2 + 1 = 3 \)
- The y-component: \( -3 + 5 = 2 \)
- The z-component: \( 4 + 2 = 6 \)

So, the resultant vector is:

\[ \overset{\rarr}{R} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \]

### Step 2: Set up the equation for the unit vector along the y-axis

We know that when \( \overset{\rarr}{A} \) is added to \( \overset{\rarr}{R} \), it yields a unit vector along the y-axis, which is represented as \( \hat{j} \).

Thus, we can write the equation:

\[ \overset{\rarr}{A} + \overset{\rarr}{R} = \hat{j} \]

Substituting \( \overset{\rarr}{R} \):

\[ \overset{\rarr}{A} + (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) = \hat{j} \]

### Step 3: Solve for \( \overset{\rarr}{A} \)

Rearranging the equation to isolate \( \overset{\rarr}{A} \):

\[ \overset{\rarr}{A} = \hat{j} - (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) \]

Distributing the negative sign:

\[ \overset{\rarr}{A} = \hat{j} - 3 \hat{i} - 2 \hat{j} - 6 \hat{k} \]

Combining like terms:

\[ \overset{\rarr}{A} = -3 \hat{i} + (1 - 2) \hat{j} - 6 \hat{k} \]

This simplifies to:

\[ \overset{\rarr}{A} = -3 \hat{i} - \hat{j} - 6 \hat{k} \]

### Final Answer

Thus, the vector \( \overset{\rarr}{A} \) is:

\[ \overset{\rarr}{A} = -3 \hat{i} - \hat{j} - 6 \hat{k} \]