To find the value of [tex]\((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\)[/tex], we proceed as follows:
1. Determine the components of [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex]:
[tex]\[
\vec{A} = 6 \vec{i} + 5 \vec{j} \quad \text{and} \quad \vec{B} = 4 \vec{i} - 3 \vec{j}
\][/tex]
2. Calculate [tex]\(\vec{A} + \vec{B}\)[/tex]:
[tex]\[
\vec{A} + \vec{B} = (6 \vec{i} + 5 \vec{j}) + (4 \vec{i} - 3 \vec{j}) = (6 + 4)\vec{i} + (5 - 3)\vec{j} = 10 \vec{i} + 2 \vec{j}
\][/tex]
3. Calculate [tex]\(\vec{A} - \vec{B}\)[/tex]:
[tex]\[
\vec{A} - \vec{B} = (6 \vec{i} + 5 \vec{j}) - (4 \vec{i} - 3 \vec{j}) = (6 - 4)\vec{i} + (5 + 3)\vec{j} = 2 \vec{i} + 8 \vec{j}
\][/tex]
4. Compute the dot product [tex]\((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\)[/tex]:
[tex]\[
(10 \vec{i} + 2 \vec{j}) \cdot (2 \vec{i} + 8 \vec{j}) = (10 \cdot 2) + (2 \cdot 8)
\][/tex]
5. Perform the multiplications:
[tex]\[
10 \cdot 2 = 20
\][/tex]
[tex]\[
2 \cdot 8 = 16
\][/tex]
6. Add the results:
[tex]\[
20 + 16 = 36
\][/tex]
Therefore, the value of [tex]\((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\)[/tex] is [tex]\(\boxed{36}\)[/tex].
