Two vectors are given by [tex]\vec{a}=-2 \hat{i}+\hat{j}-3 \hat{k}[/tex] and [tex]\vec{b}=5 \hat{i}+3 \hat{j}-2 \hat{k}[/tex]. If [tex]3 \vec{a} + 2 \vec{b} - \vec{c} = 0[/tex], then [tex]\vec{c}[/tex] is:

A. [tex]4 \hat{i} + 9 \hat{j} - 13 \hat{k}[/tex]

B. [tex]-4 \hat{i} - 9 \hat{j} + 13 \hat{k}[/tex]

C. [tex]4 \hat{i} - 9 \hat{j} - 13 \hat{k}[/tex]

D. [tex]2 \hat{i} - 3 \hat{j} + 13 \hat{k}[/tex]



Answer :

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).