The position vectors of P and Q are `5hati+4hatj+ahatk` and `-hati+2hatj-2hatk`, respectively. If the distance betwee them is 7, then the value of a will be

Options

A. `-5,1`
B. `5,1`
C. `0,5`
D. 1,0

Explanation

To find the value of \( a \) given the position vectors of points \( P \) and \( Q \) and the distance between them, we can follow these steps:

### Step 1: Write down the position vectors
The position vectors of points \( P \) and \( Q \) are given as:
\[ \vec{P} = 5\hat{i} + 4\hat{j} + a\hat{k} \]
\[ \vec{Q} = -\hat{i} + 2\hat{j} - 2\hat{k} \]

### Step 2: Find the distance formula
The distance \( d \) between two points represented by position vectors \( \vec{P} \) and \( \vec{Q} \) is given by:
\[ d = |\vec{P} - \vec{Q}| \]
This can be calculated using the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 3: Calculate the components of \( \vec{P} - \vec{Q} \)
Now, we compute \( \vec{P} - \vec{Q} \):
\[ \vec{P} - \vec{Q} = (5 - (-1))\hat{i} + (4 - 2)\hat{j} + (a - (-2))\hat{k} \]
This simplifies to:
\[ \vec{P} - \vec{Q} = (5 + 1)\hat{i} + (4 - 2)\hat{j} + (a + 2)\hat{k} \]
\[ \vec{P} - \vec{Q} = 6\hat{i} + 2\hat{j} + (a + 2)\hat{k} \]

### Step 4: Calculate the magnitude of \( \vec{P} - \vec{Q} \)
Now we find the magnitude:
\[ d = |\vec{P} - \vec{Q}| = \sqrt{(6)^2 + (2)^2 + (a + 2)^2} \]
\[ d = \sqrt{36 + 4 + (a + 2)^2} \]
\[ d = \sqrt{40 + (a + 2)^2} \]

### Step 5: Set the distance equal to 7
According to the problem, the distance \( d \) is equal to 7:
\[ \sqrt{40 + (a + 2)^2} = 7 \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 40 + (a + 2)^2 = 49 \]

### Step 7: Solve for \( (a + 2)^2 \)
Subtract 40 from both sides:
\[ (a + 2)^2 = 49 - 40 \]
\[ (a + 2)^2 = 9 \]

### Step 8: Take the square root of both sides
Taking the square root gives:
\[ a + 2 = 3 \quad \text{or} \quad a + 2 = -3 \]

### Step 9: Solve for \( a \)
1. From \( a + 2 = 3 \):
\[ a = 3 - 2 = 1 \]
2. From \( a + 2 = -3 \):
\[ a = -3 - 2 = -5 \]

### Final Answer
The possible values of \( a \) are:
\[ a = 1 \quad \text{or} \quad a = -5 \]