We are given the following:

1. $2a + b = -5\hat{j} + \hat{k}$
2. $c = \hat{i} - 2\hat{j} + 2\hat{k}$
3. $a \cdot c = 5$
4. $b$ is a unit vector, i.e., $|b| = 1$

We need to find $b \cdot c$.

Step 1: Express $b$ in terms of $a$

From the equation $2a + b = -5\hat{j} + \hat{k}$, we can solve for $b$:

$b = -5\hat{j} + \hat{k} - 2a$

Step 2: Use the dot product relation

We are given that $a \cdot c = 5$, and $c = \hat{i} - 2\hat{j} + 2\hat{k}$.

Let's find the dot product between $b$ and $c$:

$b \cdot c = (-5\hat{j} + \hat{k} - 2a) \cdot (\hat{i} - 2\hat{j} + 2\hat{k})$

Expanding this:

$b \cdot c = (-5\hat{j}) \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) + \hat{k} \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) + (-2a) \cdot (\hat{i} - 2\hat{j} + 2\hat{k})$

Now calculate the individual dot products:

1. $(-5\hat{j}) \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) = 0 - (-10) + 0 = 10$
2. $\hat{k} \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) = 0 + 0 + 2 = 2$
3. $(-2a) \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) = -2(a \cdot (\hat{i} - 2\hat{j} + 2\hat{k}))$

We already know that $a \cdot c = 5$, and $c = \hat{i} - 2\hat{j} + 2\hat{k}$. So,

$a \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) = 5$

Thus,

$(-2a) \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) = -2 \times 5 = -10$

Step 3: Combine the results

Now combine all the terms:

$b \cdot c = 10 + 2 - 10 = 2$

Final Answer:

$b \cdot c = 2$

---

Would you like to go over any details of this solution? Here are some related questions you can explore further:

1. What is the magnitude of vector $b$?
2. How can you express the vector $a$ in terms of $b$ and $c$?
3. What is the cross product of vectors $a$ and $c$?
4. If $b$ is a unit vector, how does that affect its components?
5. How would you find the angle between vectors $b$ and $c$?

Tip: When working with vectors, remember to always break them down into components for easier manipulation when performing operations like dot products or cross products!