Let's analyze the problem step-by-step. We are given that the angle between vectors [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex] is [tex]\(30^\circ\)[/tex]. We need to determine whether each statement (A, B, C, and D) is correct.
### Statement A: Angle between [tex]\(\hat{A}\)[/tex] and [tex]\(-\hat{B}\)[/tex]
Here, [tex]\(\hat{A}\)[/tex] and [tex]\(-\hat{B}\)[/tex] represent the unit vectors in the direction of [tex]\(\vec{A}\)[/tex] and [tex]\(-\vec{B}\)[/tex], respectively. Since the direction of [tex]\(-\hat{B}\)[/tex] is opposite to that of [tex]\(\hat{B}\)[/tex], the angle between [tex]\(\hat{A}\)[/tex] and [tex]\(-\hat{B}\)[/tex] would be:
[tex]\[ 180^\circ - 30^\circ = 150^\circ \][/tex]
Indeed, the angle between [tex]\(\hat{A}\)[/tex] and [tex]\(-\hat{B}\)[/tex] is [tex]\(150^\circ\)[/tex].
### Statement B: Angle between [tex]\(2 \vec{A}\)[/tex] and [tex]\(3 \vec{B}\)[/tex]
Scaling vectors [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex] by any scalar values (in this case, 2 and 3, respectively) does not change the angle between them. Therefore, the angle between [tex]\(2 \vec{A}\)[/tex] and [tex]\(3 \vec{B}\)[/tex] remains [tex]\(30^\circ\)[/tex].
### Statements C and D: Angle between the resultant of [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex] and [tex]\(-\vec{B}\)[/tex]
First, let's denote the resultant vector of [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex] as [tex]\(\vec{R}\)[/tex]:
[tex]\[ \vec{R} = \vec{A} + \vec{B} \][/tex]
To find the angle between [tex]\(\vec{R}\)[/tex] and [tex]\(-\vec{B}\)[/tex], we need more specific information about the resultant and angles. However, considering potential outcomes for these vectors, it's possible that the resultant vector [tex]\(\vec{R}\)[/tex] forms an angle which makes the resultant angle to [tex]\(-\vec{B}\)[/tex]:
It might align in such a way that it's not a simple deduction. One case provided:
[tex]\[ \text{Angle between } \vec{R} \text{ and } -\vec{B} \text{ is } 200^\circ \][/tex]
Another case given:
[tex]\[ \text{Angle between } \vec{R} \text{ and } -\vec{B} \text{ is } 175^\circ \][/tex]
Given specific resultant vectors and orientations in space, these might be possible, depending on how vectors [tex]\(\vec{A}\)[/tex] and [tex]\(\vec{B}\)[/tex] add and their resultant vector direction.
### Conclusion:
- A) Correct. Angle between [tex]\(\hat{A}\)[/tex] and [tex]\(-\hat{B}\)[/tex] is [tex]\(150^\circ\)[/tex].
- B) Correct. Angle between [tex]\(2 \vec{A}\)[/tex] and [tex]\(3 \vec{B}\)[/tex] is [tex]\(30^\circ\)[/tex].
- C & D) Given possibilities, they both can be considered since exact resultant calculation not specified might lead to such resultant angle possibilities.
Thus, statements (A), (B), (C), and (D) can be valid conclusions based on how angles are computed and possible resultant placements.
