Let `vec a=3 hat i+hat j-2 hat k,vec b=4 hat i+hat j+7 hat k` and `vec c= hat i-3 hat j+4 hat k` be three vectors.If a vectors `vec p` satisfies `vec p timesvec b=vec c timesvec b` and `vec p*vec a=0` ,then `vec p*(hat i-hat j-hat k)` is equal to

Options

A. 28
B. 32
C. 24
D. 36

Answer

To solve the problem, we need to find the vector \(\vec{p}\) that satisfies the conditions given in the question. Let's break down the solution step by step.

### Given:
- \(\vec{a} = 3\hat{i} + \hat{j} - 2\hat{k}\)
- \(\vec{b} = 4\hat{i} + \hat{j} + 7\hat{k}\)
- \(\vec{c} = \hat{i} - 3\hat{j} + 4\hat{k}\)

### Conditions:
1. \(\vec{p} \times \vec{b} = \vec{c} \times \vec{b}\)
2. \(\vec{p} \cdot \vec{a} = 0\)

### Step 1: Use the first condition
From the first condition, we can rewrite it as:
\[ \vec{p} \times \vec{b} - \vec{c} \times \vec{b} = 0 \]
This implies that:
\[ \vec{p} \times \vec{b} = \vec{c} \times \vec{b} \]

### Step 2: Rearranging the equation
This means that \(\vec{p} - \vec{c}\) is parallel to \(\vec{b}\):
\[ \vec{p} - \vec{c} = \lambda \vec{b} \quad \text{for some scalar } \lambda \]
Thus, we can express \(\vec{p}\) as:
\[ \vec{p} = \vec{c} + \lambda \vec{b} \]

### Step 3: Use the second condition
Now, we substitute \(\vec{p}\) into the second condition:
\[ (\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 0 \]
Expanding this gives:
\[ \vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0 \]

### Step 4: Calculate the dot products
Now we calculate \(\vec{c} \cdot \vec{a}\) and \(\vec{b} \cdot \vec{a}\):
- \(\vec{c} \cdot \vec{a} = (1)(3) + (-3)(1) + (4)(-2) = 3 - 3 - 8 = -8\)
- \(\vec{b} \cdot \vec{a} = (4)(3) + (1)(1) + (7)(-2) = 12 + 1 - 14 = -1\)

### Step 5: Substitute the dot products back
Substituting these values back into our equation:
\[ -8 + \lambda (-1) = 0 \]
This simplifies to:
\[ \lambda = 8 \]

### Step 6: Find \(\vec{p}\)
Now we can substitute \(\lambda\) back into the equation for \(\vec{p}\):
\[ \vec{p} = \vec{c} + 8\vec{b} = (\hat{i} - 3\hat{j} + 4\hat{k}) + 8(4\hat{i} + \hat{j} + 7\hat{k}) \]
Calculating this gives:
\[ \vec{p} = \hat{i} - 3\hat{j} + 4\hat{k} + (32\hat{i} + 8\hat{j} + 56\hat{k}) = (1 + 32)\hat{i} + (-3 + 8)\hat{j} + (4 + 56)\hat{k} \]
Thus,
\[ \vec{p} = 33\hat{i} + 5\hat{j} + 60\hat{k} \]

### Step 7: Calculate \(\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})\)
Now we need to find:
\[ \vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) = (33)(1) + (5)(-1) + (60)(-1) \]
Calculating this gives:
\[ 33 - 5 - 60 = 33 - 65 = -32 \]

### Final Answer
Thus, the value of \(\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})\) is:
\[ \boxed{-32} \]