Are the following set of vectors linearly independent?
`vec(a) = hat(i) - 2hat(j) + 3hat(k), vec(b) = 3hat(j) - 6hat(j) + 9hat(k)`

Answer

To determine whether the vectors \(\vec{a}\) and \(\vec{b}\) are linearly independent, we will follow these steps:

### Step 1: Write the vectors in component form
The vectors are given as:
\[ \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \]
\[ \vec{b} = 3\hat{i} - 6\hat{j} + 9\hat{k} \]

### Step 2: Check if one vector is a scalar multiple of the other
We will check if there exists a scalar \(k\) such that:
\[ \vec{b} = k \cdot \vec{a} \]

### Step 3: Set up the equations
From the components of the vectors, we can set up the following equations:
1. \(3 = k \cdot 1\) (from the \(\hat{i}\) components)
2. \(-6 = k \cdot (-2)\) (from the \(\hat{j}\) components)
3. \(9 = k \cdot 3\) (from the \(\hat{k}\) components)

### Step 4: Solve for \(k\)
From the first equation:
\[ k = 3 \]
From the second equation:
\[ -6 = k \cdot (-2) \implies k = \frac{-6}{-2} = 3 \]
From the third equation:
\[ 9 = k \cdot 3 \implies k = \frac{9}{3} = 3 \]

### Step 5: Conclusion
Since we found the same scalar \(k = 3\) for all three components, we can conclude that:
\[ \vec{b} = 3 \cdot \vec{a} \]
This means that \(\vec{b}\) is a scalar multiple of \(\vec{a}\), indicating that the two vectors are linearly dependent.

### Final Answer
The vectors \(\vec{a}\) and \(\vec{b}\) are not linearly independent; they are linearly dependent.

---