If p, q, r and s be respectively the magnitude of the vectors `3hat(i)-2hat(j), 2hat(i) +2hat(j) + hat(k), 4hat(i)- hat(j) + hat(k), 2hat(i) + 2hat(j) + 3hat(k)`. Then, which one of the following is correct?

Options

A. `r gt s gt q gt p`
B. `s gt r gt p gt q`
C. `r gt s gt p gt q`
D. `s gt r gt q gt p`

Answer

To solve the problem, we need to find the magnitudes of the given vectors \( p, q, r, \) and \( s \). Let's calculate each magnitude step by step.

### Step 1: Calculate the magnitude of vector \( p \)
The vector \( p \) is given as:
\[ p = 3\hat{i} - 2\hat{j} \]
The magnitude of a vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is calculated using the formula:
\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]
For vector \( p \):
- \( a = 3 \)
- \( b = -2 \)
- \( c = 0 \)

Calculating the magnitude:
\[ |p| = \sqrt{3^2 + (-2)^2 + 0^2} = \sqrt{9 + 4 + 0} = \sqrt{13} \]

### Step 2: Calculate the magnitude of vector \( q \)
The vector \( q \) is given as:
\[ q = 2\hat{i} + 2\hat{j} + \hat{k} \]
Calculating the magnitude:
- \( a = 2 \)
- \( b = 2 \)
- \( c = 1 \)

\[ |q| = \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 3: Calculate the magnitude of vector \( r \)
The vector \( r \) is given as:
\[ r = 4\hat{i} - \hat{j} + \hat{k} \]
Calculating the magnitude:
- \( a = 4 \)
- \( b = -1 \)
- \( c = 1 \)

\[ |r| = \sqrt{4^2 + (-1)^2 + 1^2} = \sqrt{16 + 1 + 1} = \sqrt{18} \]

### Step 4: Calculate the magnitude of vector \( s \)
The vector \( s \) is given as:
\[ s = 2\hat{i} + 2\hat{j} + 3\hat{k} \]
Calculating the magnitude:
- \( a = 2 \)
- \( b = 2 \)
- \( c = 3 \)

\[ |s| = \sqrt{2^2 + 2^2 + 3^2} = \sqrt{4 + 4 + 9} = \sqrt{17} \]

### Step 5: Compare the magnitudes
Now we have the magnitudes:
- \( |p| = \sqrt{13} \)
- \( |q| = 3 \)
- \( |r| = \sqrt{18} \)
- \( |s| = \sqrt{17} \)

To compare these, we can approximate the square roots:
- \( \sqrt{13} \approx 3.6 \)
- \( 3 = 3 \)
- \( \sqrt{18} \approx 4.24 \)
- \( \sqrt{17} \approx 4.12 \)

Ordering the magnitudes:
1. \( |r| = \sqrt{18} \) (largest)
2. \( |s| = \sqrt{17} \)
3. \( |p| = \sqrt{13} \)
4. \( |q| = 3 \) (smallest)

Thus, the order of the magnitudes is:
\[ r > s > p > q \]

### Conclusion
The correct answer is \( r > s > p > q \).