To determine \(\vec{c}\) in the equation \(3 \vec{a} + 2 \vec{b} - \vec{c} = 0\), we can follow these steps:
1. Express the vectors \(\vec{a}\) and \(\vec{b}\) in component form:
[tex]\[
\vec{a} = -2 \hat{i} + \hat{j} - 3 \hat{k}
\][/tex]
[tex]\[
\vec{b} = 5 \hat{i} + 3 \hat{j} - 2 \hat{k}
\][/tex]
2. Multiply \(\vec{a}\) by 3 and \(\vec{b}\) by 2:
[tex]\[
3 \vec{a} = 3(-2 \hat{i} + \hat{j} - 3 \hat{k}) = -6 \hat{i} + 3 \hat{j} - 9 \hat{k}
\][/tex]
[tex]\[
2 \vec{b} = 2(5 \hat{i} + 3 \hat{j} - 2 \hat{k}) = 10 \hat{i} + 6 \hat{j} - 4 \hat{k}
\][/tex]
3. Add the scaled vectors \(3 \vec{a}\) and \(2 \vec{b}\):
[tex]\[
3 \vec{a} + 2 \vec{b} = (-6 \hat{i} + 3 \hat{j} - 9 \hat{k}) + (10 \hat{i} + 6 \hat{j} - 4 \hat{k})
\][/tex]
[tex]\[
3 \vec{a} + 2 \vec{b} = (-6 + 10) \hat{i} + (3 + 6) \hat{j} + (-9 - 4) \hat{k}
\][/tex]
[tex]\[
3 \vec{a} + 2 \vec{b} = 4 \hat{i} + 9 \hat{j} - 13 \hat{k}
\][/tex]
4. Rearrange the original equation and solve for \(\vec{c}\):
[tex]\[
3 \vec{a} + 2 \vec{b} - \vec{c} = 0
\][/tex]
[tex]\[
\vec{c} = 3 \vec{a} + 2 \vec{b}
\][/tex]
Substituting the result from step 3:
[tex]\[
\vec{c} = 4 \hat{i} + 9 \hat{j} - 13 \hat{k}
\][/tex]
Therefore, the correct value for \(\vec{c}\) is:
1) \(4 \hat{i} + 9 \hat{j} - 13 \hat{k}\).
The answer is choice (1).
