Consider three vectors `p=hat(i)+hat(j)+hat(k), q=2hat(i)+4hat(j)-hat(k) and r=hat(i)+hat(j)+3hat(k)` and let s be a unit vector, then Q. The magnitude of the vector `(p*s)(qtimesr)+(q*s)(r xx p)+(r*s)(ptimesq)` is

Options

A. A. `4`
B. B. `8`
C. C. `18`
D. D. `2`

Answer

To solve the problem, we need to find the magnitude of the vector \( (p \cdot s)(q \times r) + (q \cdot s)(r \times p) + (r \cdot s)(p \times q) \) where \( p, q, r \) are given vectors and \( s \) is a unit vector.

### Step 1: Define the vectors
Given:
- \( p = \hat{i} + \hat{j} + \hat{k} \)
- \( q = 2\hat{i} + 4\hat{j} - \hat{k} \)
- \( r = \hat{i} + \hat{j} + 3\hat{k} \)

### Step 2: Calculate the cross products
1. **Calculate \( q \times r \)**:
\[ q \times r = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & -1 \\ 1 & 1 & 3 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i} \begin{vmatrix} 4 & -1 \\ 1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & -1 \\ 1 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 4 \\ 1 & 1 \end{vmatrix} \]
\[ = \hat{i} (4 \cdot 3 - (-1) \cdot 1) - \hat{j} (2 \cdot 3 - (-1) \cdot 1) + \hat{k} (2 \cdot 1 - 4 \cdot 1) \]
\[ = \hat{i} (12 + 1) - \hat{j} (6 + 1) + \hat{k} (2 - 4) \]
\[ = 13\hat{i} - 7\hat{j} - 2\hat{k} \]

2. **Calculate \( r \times p \)**:
\[ r \times p = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 3 \\ 1 & 1 & 1 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i} \begin{vmatrix} 1 & 3 \\ 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} \]
\[ = \hat{i} (1 \cdot 1 - 3 \cdot 1) - \hat{j} (1 \cdot 1 - 3 \cdot 1) + \hat{k} (1 \cdot 1 - 1 \cdot 1) \]
\[ = \hat{i} (1 - 3) - \hat{j} (1 - 3) + 0\hat{k} \]
\[ = -2\hat{i} + 2\hat{j} \]

3. **Calculate \( p \times q \)**:
\[ p \times q = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 2 & 4 & -1 \end{vmatrix} \]
Expanding this determinant:
\[ = \hat{i} \begin{vmatrix} 1 & 1 \\ 4 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 2 & 4 \end{vmatrix} \]
\[ = \hat{i} (1 \cdot (-1) - 1 \cdot 4) - \hat{j} (1 \cdot (-1) - 1 \cdot 2) + \hat{k} (1 \cdot 4 - 1 \cdot 2) \]
\[ = \hat{i} (-1 - 4) - \hat{j} (-1 - 2) + \hat{k} (4 - 2) \]
\[ = -5\hat{i} + 3\hat{j} + 2\hat{k} \]

### Step 3: Calculate the triple product \( p \cdot (q \times r) \)
Using the result from \( q \times r \):
\[ p \cdot (q \times r) = \hat{i} \cdot (13\hat{i} - 7\hat{j} - 2\hat{k}) + \hat{j} \cdot (13\hat{i} - 7\hat{j} - 2\hat{k}) + \hat{k} \cdot (13\hat{i} - 7\hat{j} - 2\hat{k}) \]
\[ = 1 \cdot 13 + 1 \cdot (-7) + 1 \cdot (-2) = 13 - 7 - 2 = 4 \]

### Step 4: Calculate the triple product \( q \cdot (r \times p) \)
Using the result from \( r \times p \):
\[ q \cdot (r \times p) = (2\hat{i} + 4\hat{j} - \hat{k}) \cdot (-2\hat{i} + 2\hat{j}) = 2 \cdot (-2) + 4 \cdot 2 + (-1) \cdot 0 \]
\[ = -4 + 8 = 4 \]

### Step 5: Calculate the triple product \( r \cdot (p \times q) \)
Using the result from \( p \times q \):
\[ r \cdot (p \times q) = (\hat{i} + \hat{j} + 3\hat{k}) \cdot (-5\hat{i} + 3\hat{j} + 2\hat{k}) = 1 \cdot (-5) + 1 \cdot 3 + 3 \cdot 2 \]
\[ = -5 + 3 + 6 = 4 \]

### Step 6: Combine the results
Now we can combine the results:
\[ (p \cdot s)(q \times r) + (q \cdot s)(r \times p) + (r \cdot s)(p \times q) = 4 + 4 + 4 = 12 \]

### Step 7: Find the magnitude
Since \( s \) is a unit vector, the magnitude of the vector is simply the sum of the coefficients:
\[ \text{Magnitude} = |12| = 12 \]

### Final Answer
The magnitude of the vector is \( 12 \).