Two vectors are given by `veca=-2hati+hatj-3hatk and vecb=5hati+3hatj-2hatk`. If `3veca+2vecb-vec c=0` then third vector `vecc` is

Options

A. `4hati+9hatj-13hatk`
B. `-4hati-9hatj+13hatk`
C. `4hati-9hatk-13hatk`
D. `2hati-3hatj-13hatk`

Answer

To solve the problem, we need to find the vector \(\vec{c}\) given the equation \(3\vec{a} + 2\vec{b} - \vec{c} = 0\).

### Step-by-Step Solution:

1. **Write down the vectors**:
\[ \vec{a} = -2\hat{i} + \hat{j} - 3\hat{k} \]
\[ \vec{b} = 5\hat{i} + 3\hat{j} - 2\hat{k} \]

2. **Rearrange the equation**:
From the equation \(3\vec{a} + 2\vec{b} - \vec{c} = 0\), we can express \(\vec{c}\) as:
\[ \vec{c} = 3\vec{a} + 2\vec{b} \]

3. **Calculate \(3\vec{a}\)**:
\[ 3\vec{a} = 3(-2\hat{i} + \hat{j} - 3\hat{k}) = -6\hat{i} + 3\hat{j} - 9\hat{k} \]

4. **Calculate \(2\vec{b}\)**:
\[ 2\vec{b} = 2(5\hat{i} + 3\hat{j} - 2\hat{k}) = 10\hat{i} + 6\hat{j} - 4\hat{k} \]

5. **Add \(3\vec{a}\) and \(2\vec{b}\)**:
Now we add the results from steps 3 and 4:
\[ \vec{c} = (-6\hat{i} + 3\hat{j} - 9\hat{k}) + (10\hat{i} + 6\hat{j} - 4\hat{k}) \]

Combine the components:
- For \(\hat{i}\):
\[ -6 + 10 = 4\hat{i} \]
- For \(\hat{j}\):
\[ 3 + 6 = 9\hat{j} \]
- For \(\hat{k}\):
\[ -9 - 4 = -13\hat{k} \]

6. **Final result**:
Therefore, the vector \(\vec{c}\) is:
\[ \vec{c} = 4\hat{i} + 9\hat{j} - 13\hat{k} \]

### Final Answer:
\[ \vec{c} = 4\hat{i} + 9\hat{j} - 13\hat{k} \]