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

Explanation

To solve the problem step by step, we start with the given vectors and conditions:

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

We need to find the vector \(\vec{r}\) such that:
1. \(\vec{r} \times \vec{b} = \vec{c} \times \vec{b}\)
2. \(\vec{r} \cdot \vec{a} = 0\)

Finally, we need to find the value of \(\vec{r} \cdot \vec{b}\).

### Step 1: Use the cross product condition

From the condition \(\vec{r} \times \vec{b} = \vec{c} \times \vec{b}\), we can take the cross product with \(\vec{a}\) on both sides:

\[ \vec{a} \times (\vec{r} \times \vec{b}) = \vec{a} \times (\vec{c} \times \vec{b}) \]

Using the vector triple product identity, we have:

\[ \vec{a} \times (\vec{r} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{r} - (\vec{a} \cdot \vec{r}) \vec{b} \]

And similarly for the right-hand side:

\[ \vec{a} \times (\vec{c} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{c} - (\vec{a} \cdot \vec{c}) \vec{b} \]

### Step 2: Calculate \(\vec{a} \cdot \vec{b}\) and \(\vec{a} \cdot \vec{c}\)

Now we need to compute \(\vec{a} \cdot \vec{b}\):

\[ \vec{a} \cdot \vec{b} = (-1)(-1) + (0)(1) + (-1)(0) = 1 \]

Next, calculate \(\vec{a} \cdot \vec{c}\):

\[ \vec{a} \cdot \vec{c} = (-1)(1) + (0)(2) + (-1)(3) = -1 - 3 = -4 \]

### Step 3: Substitute into the equation

Now substituting these values into our equation:

\[ 1 \vec{r} - 0 \vec{b} = 1 \vec{c} - (-4) \vec{b} \]

This simplifies to:

\[ \vec{r} = \vec{c} + 4\vec{b} \]

### Step 4: Substitute the vectors

Substituting the values of \(\vec{c}\) and \(\vec{b}\):

\[ \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + 4(-\hat{i} + \hat{j}) \]

Calculating this gives:

\[ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} - 4\hat{i} + 4\hat{j} = (-3\hat{i} + 6\hat{j} + 3\hat{k}) \]

### Step 5: Calculate \(\vec{r} \cdot \vec{b}\)

Now we need to find \(\vec{r} \cdot \vec{b}\):

\[ \vec{r} \cdot \vec{b} = (-3\hat{i} + 6\hat{j} + 3\hat{k}) \cdot (-\hat{i} + \hat{j}) \]

Calculating this gives:

\[ = (-3)(-1) + (6)(1) + (3)(0) = 3 + 6 + 0 = 9 \]

### Final Answer

Thus, the value of \(\vec{r} \cdot \vec{b}\) is:

\[ \boxed{9} \]