Let's solve the problem step by step.

Given:

• $\mathbf{a} = \hat{\mathbf{j}} - \hat{\mathbf{k}}$
• $\mathbf{c} = \hat{\mathbf{i}} - \hat{\mathbf{j}} - \hat{\mathbf{k}}$
• The conditions:
  1. $\mathbf{a} \times \mathbf{b} + \mathbf{c} = \mathbf{0}$
  2. $\mathbf{a} \cdot \mathbf{b} = 3$

Step 1: Determine $\mathbf{b}$ from the cross product condition

The first condition $\mathbf{a} \times \mathbf{b} + \mathbf{c} = \mathbf{0}$ implies: $\mathbf{a} \times \mathbf{b} = -\mathbf{c}$

Step 2: Compute the cross product $\mathbf{a} \times \mathbf{b}$

Let $\mathbf{b} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$.

The cross product $\mathbf{a} \times \mathbf{b}$ is calculated as:

\begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 0 & 1 & -1 \\ x & y & z \\ \end{vmatrix} = \hat{\mathbf{i}} \left(1z - (-1)y\right) - \hat{\mathbf{j}}(0z - (-1)x) + \hat{\mathbf{k}}(0y - 1x)$$ $$\mathbf{a} \times \mathbf{b} = (z + y)\hat{\mathbf{i}} - x\hat{\mathbf{j}} - x\hat{\mathbf{k}}$$ We know $$\mathbf{a} \times \mathbf{b} = -\mathbf{c}$$, so: $$(z + y)\hat{\mathbf{i}} - x\hat{\mathbf{j}} - x\hat{\mathbf{k}} = -(\hat{\mathbf{i}} - \hat{\mathbf{j}} - \hat{\mathbf{k}})$$ This gives us the system of equations: 1. $$z + y = -1$$ 2. $$-x = 1$$ 3. $$-x = 1$$ From (2) and (3), $$x = -1$$. From (1), $$y + z = -1$$. ### Step 3: Use the dot product condition The second condition $$\mathbf{a} \cdot \mathbf{b} = 3$$ is: $$\mathbf{a} \cdot \mathbf{b} = y - z = 3$$ We now have two equations: 1. $$y + z = -1$$ 2. $$y - z = 3$$ Adding these equations: $$2y = 2 \quad \Rightarrow \quad y = 1$$ Subtracting them: $$2z = -4 \quad \Rightarrow \quad z = -2$$ ### Step 4: Write the vector $$\mathbf{b}$$ $$\mathbf{b} = -\hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}$$ This corresponds to option **(c)**: $$\mathbf{b} = -\hat{\mathbf{i}} + \hat{\mathbf{j}} - 2\hat{\mathbf{k}}$$ ### Final Answer: The correct answer is $$\boxed{c}$$. Would you like more details, or do you have any questions? Here are some related questions: 1. What is the geometric interpretation of the cross product condition? 2. How do you solve a system of linear equations like in this problem? 3. What are some properties of the dot product that are used in vector operations? 4. How does the direction of a vector affect its cross product with another vector? 5. Can you derive the general formula for the cross product of two vectors? **Tip:** When working with vectors, always consider the direction and magnitude, as they are crucial in vector operations like dot and cross products.