Let `veca=-hati-hatk, vecb=-hati+hatj` and `vecc=hati+2hatj+3hatk`
be three given vectors. If `vecr` is a vector such that `vecrxxvecb=veccxxvecb` and `vecr.veca=0`, then the value of `vecr.vecb` is

Options

A. `4`
B. `8`
C. `6`
D. `9`

Explanation

To solve the problem, we need to find the value of \( \vec{r} \cdot \vec{b} \) given the conditions \( \vec{r} \times \vec{b} = \vec{c} \times \vec{b} \) and \( \vec{r} \cdot \vec{a} = 0 \).

### Step 1: Write down the vectors
We have:
\[ \vec{a} = -\hat{i} - \hat{k}, \quad \vec{b} = -\hat{i} + \hat{j}, \quad \vec{c} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Use the cross product condition
From the condition \( \vec{r} \times \vec{b} = \vec{c} \times \vec{b} \), we can rearrange it to:
\[ \vec{r} \times \vec{b} - \vec{c} \times \vec{b} = 0 \]
This implies that \( \vec{r} - \vec{c} \) is parallel to \( \vec{b} \). Therefore, we can express \( \vec{r} \) as:
\[ \vec{r} = \vec{c} + \lambda \vec{b} \]
where \( \lambda \) is a scalar.

### Step 3: Apply the dot product condition
Now, we also have the condition \( \vec{r} \cdot \vec{a} = 0 \):
\[ (\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 0 \]
Expanding this gives:
\[ \vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0 \]

### Step 4: Calculate \( \vec{c} \cdot \vec{a} \) and \( \vec{b} \cdot \vec{a} \)
First, we calculate \( \vec{c} \cdot \vec{a} \):
\[ \vec{c} \cdot \vec{a} = (\hat{i} + 2\hat{j} + 3\hat{k}) \cdot (-\hat{i} - \hat{k}) = -1 \cdot 1 + 2 \cdot 0 + 3 \cdot (-1) = -1 - 3 = -4 \]

Next, we calculate \( \vec{b} \cdot \vec{a} \):
\[ \vec{b} \cdot \vec{a} = (-\hat{i} + \hat{j}) \cdot (-\hat{i} - \hat{k}) = 1 \cdot 1 + 0 \cdot 1 + 0 \cdot (-1) = 1 \]

### Step 5: Substitute back to find \( \lambda \)
Substituting these values into the equation gives:
\[ -4 + \lambda (1) = 0 \implies \lambda = 4 \]

### Step 6: Substitute \( \lambda \) back into the expression for \( \vec{r} \)
Now we substitute \( \lambda \) back into the expression for \( \vec{r} \):
\[ \vec{r} = \vec{c} + 4\vec{b} = (\hat{i} + 2\hat{j} + 3\hat{k}) + 4(-\hat{i} + \hat{j}) = \hat{i} + 2\hat{j} + 3\hat{k} - 4\hat{i} + 4\hat{j} = -3\hat{i} + 6\hat{j} + 3\hat{k} \]

### Step 7: Calculate \( \vec{r} \cdot \vec{b} \)
Now we can find \( \vec{r} \cdot \vec{b} \):
\[ \vec{r} \cdot \vec{b} = (-3\hat{i} + 6\hat{j} + 3\hat{k}) \cdot (-\hat{i} + \hat{j}) = 3 \cdot 1 + 6 \cdot 1 + 0 = 3 + 6 = 9 \]

### Final Answer
Thus, the value of \( \vec{r} \cdot \vec{b} \) is:
\[ \boxed{9} \]