To solve for [tex]\( u \cdot v \)[/tex], where [tex]\( u = 1 + i \sqrt{3} \)[/tex] and [tex]\( v = 1 - i \sqrt{3} \)[/tex], follow these steps:
1. Express the problem:
We need to find the product of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[
u \cdot v = (1 + i \sqrt{3})(1 - i \sqrt{3})
\][/tex]
2. Use the difference of squares formula:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = i\sqrt{3} \)[/tex].
3. Apply the formula to our specific terms:
[tex]\[
(1 + i \sqrt{3})(1 - i \sqrt{3}) = 1^2 - (i \sqrt{3})^2
\][/tex]
4. Compute each term:
- [tex]\( 1^2 = 1 \)[/tex]
- [tex]\((i \sqrt{3})^2 = i^2 \cdot (\sqrt{3})^2 = -1 \cdot 3 = -3 \)[/tex] because [tex]\( i^2 = -1 \)[/tex].
5. Subtract the results:
[tex]\[
1 - (-3) = 1 + 3 = 4
\][/tex]
Therefore, the product [tex]\( u \cdot v \)[/tex] is [tex]\( 4 \)[/tex]. Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
