To determine the expression that represents [tex]\(|r - s|\)[/tex], we use the Law of Cosines for vectors. Given the magnitudes and angles of vectors [tex]\(r\)[/tex] and [tex]\(s\)[/tex]:
- The magnitude of [tex]\(r\)[/tex] is 6 and its angle from the positive x-axis is [tex]\(30^\circ\)[/tex].
- The magnitude of [tex]\(s\)[/tex] is 11 and its angle from the positive x-axis is [tex]\(225^\circ\)[/tex].
We'll follow these steps:
1. Calculate the angle difference between [tex]\(r\)[/tex] and [tex]\(s\)[/tex]:
The angles are [tex]\(30^\circ\)[/tex] and [tex]\(225^\circ\)[/tex], respectively. To find the angle difference, we compute:
[tex]\[
|30^\circ - 225^\circ| = | -195^\circ | = 195^\circ
\][/tex]
2. Apply the Law of Cosines:
The Law of Cosines in the context of vectors states that:
[tex]\[
|r - s| = \sqrt{ r^2 + s^2 - 2 \cdot r \cdot s \cdot \cos(\theta) }
\][/tex]
Here, [tex]\(r = 6\)[/tex], [tex]\(s = 11\)[/tex], and [tex]\(\theta\)[/tex] is the angle difference calculated, which is [tex]\(195^\circ\)[/tex]. Thus, we substitute:
[tex]\[
|r - s| = \sqrt{ 6^2 + 11^2 - 2 \cdot 6 \cdot 11 \cdot \cos(195^\circ) }
\][/tex]
Reviewing the given potential answers, the correct choice must be:
[tex]\[
\sqrt{6^2 + 11^2 - 2 \cdot 6 \cdot 11 \cos(195^\circ)}
\][/tex]
So the expression among the choices given that represents [tex]\(|r - s|\)[/tex] is:
[tex]\[
\sqrt{ 6^2 + 11^2 - 2 \cdot 6 \cdot 11 \cdot \cos(195^\circ) }
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\sqrt{6^2+11^2-2(6)(11) \cos \left(165^{\circ}\right)}}
\][/tex]
