P,Q,R,S are the points `(1,2,-2), (8,10,11), (1,2,3) and (3,5,7)` respectively. If s denotes the projection of PQ on RS then `29s^(2) + 29` is equal to :

Options

A. 2195
B. 8129
C. 3100
D. 2500

Answer

To solve the problem, we need to find the projection of vector PQ onto vector RS and then compute the expression \(29s^2 + 29\).

### Step 1: Identify the points and vectors
Given points:
- \( P(1, 2, -2) \)
- \( Q(8, 10, 11) \)
- \( R(1, 2, 3) \)
- \( S(3, 5, 7) \)

We need to find the vectors \( \vec{PQ} \) and \( \vec{RS} \).

### Step 2: Calculate vector \( \vec{PQ} \)
Using the formula for the vector between two points:
\[ \vec{PQ} = Q - P = (8 - 1) \hat{i} + (10 - 2) \hat{j} + (11 - (-2)) \hat{k} \]
Calculating each component:
\[ \vec{PQ} = 7 \hat{i} + 8 \hat{j} + 13 \hat{k} \]

### Step 3: Calculate vector \( \vec{RS} \)
Similarly, calculate vector \( \vec{RS} \):
\[ \vec{RS} = S - R = (3 - 1) \hat{i} + (5 - 2) \hat{j} + (7 - 3) \hat{k} \]
Calculating each component:
\[ \vec{RS} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]

### Step 4: Calculate the dot product \( \vec{PQ} \cdot \vec{RS} \)
Now we find the dot product:
\[ \vec{PQ} \cdot \vec{RS} = (7)(2) + (8)(3) + (13)(4) \]
Calculating:
\[ \vec{PQ} \cdot \vec{RS} = 14 + 24 + 52 = 90 \]

### Step 5: Calculate the magnitude of vector \( \vec{RS} \)
The magnitude of \( \vec{RS} \) is given by:
\[ |\vec{RS}| = \sqrt{(2^2) + (3^2) + (4^2)} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

### Step 6: Calculate the projection \( s \)
The projection \( s \) of \( \vec{PQ} \) onto \( \vec{RS} \) is given by:
\[ s = \frac{\vec{PQ} \cdot \vec{RS}}{|\vec{RS}|} = \frac{90}{\sqrt{29}} \]

### Step 7: Calculate \( s^2 \)
Now, we calculate \( s^2 \):
\[ s^2 = \left(\frac{90}{\sqrt{29}}\right)^2 = \frac{8100}{29} \]

### Step 8: Calculate \( 29s^2 + 29 \)
Now we compute \( 29s^2 + 29 \):
\[ 29s^2 + 29 = 29 \left(\frac{8100}{29}\right) + 29 = 8100 + 29 = 8129 \]

### Final Answer
Thus, the value of \( 29s^2 + 29 \) is \( \boxed{8129} \).