26. Let ˆ ˆ a j k   and ˆ ˆ ˆ c i j k     . Then the vec
tor b
 satisfying 0 a b c   
   and . 3 ab
  is
(a) ˆ ˆ ˆ 2 i j k    (b) ˆ ˆ ˆ 2 2 i j k  
(c) ˆ ˆ ˆ 2 i j k   (d) ˆ ˆ ˆ 2 i j

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:
\[
\mathbf{a} \times \mathbf{b} = 
\begin{vmatrix}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
0 & 1 & -1 \\
x & y & z \\
\end{vmatrix}
= \hat{\mathbf{i}} \left(1z - (-1)y\right) - \hat{\mathbf{j}}(0z - (-1)x) + \hat{\mathbf{k}}(0y - 1x)
\]
\[
\mathbf{a} \times \mathbf{b} = (z + y)\hat{\mathbf{i}} - x\hat{\mathbf{j}} - x\hat{\mathbf{k}}
\]

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.

Learn how to solve a vector problem involving the cross product and dot product. This problem requires understanding vector operations and solving systems of linear equations. Suitable for university-level mathematics.

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