Find the best solutions to your problems with the help of IDNLearn.com's experts. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.
Sagot :
To determine the image of point [tex]\( P \)[/tex] after it has been rotated about the origin [tex]\((0,0)\)[/tex] by [tex]\(-15^\circ\)[/tex], we follow these steps:
1. Identify the coordinates of point [tex]\( P \)[/tex] before rotation: Given point [tex]\( P \)[/tex] is at the origin, its coordinates are [tex]\( (0, 0) \)[/tex].
2. Define the rotation angle: The rotation angle is [tex]\(-15^\circ\)[/tex].
3. Construct the rotation matrix: Rotation matrices are used to handle transformations involving rotations about the origin. The general form of the rotation matrix for an angle [tex]\(\theta\)[/tex] is:
[tex]\[ R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \][/tex]
Here, [tex]\(\theta\)[/tex] is [tex]\(-15^\circ\)[/tex].
4. Apply the rotation matrix to point [tex]\( P \)[/tex]: To find the new position of point [tex]\( P \)[/tex], we multiply the rotation matrix by the coordinate vector of [tex]\( P \)[/tex].
5. Compute the new coordinates: By multiplying the rotation matrix by the coordinates [tex]\((0, 0)\)[/tex], we get:
[tex]\[ \begin{pmatrix} \cos(-15^\circ) & -\sin(-15^\circ) \\ \sin(-15^\circ) & \cos(-15^\circ) \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot \cos(-15^\circ) + 0 \cdot -\sin(-15^\circ) \\ 0 \cdot \sin(-15^\circ) + 0 \cdot \cos(-15^\circ) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \][/tex]
Thus, rotating the point [tex]\((0, 0)\)[/tex] by [tex]\(-15^\circ\)[/tex] does not change its position. The coordinates of the image of point [tex]\( P \)[/tex] remain [tex]\((0, 0)\)[/tex].
So, the image of [tex]\( P \)[/tex] after rotation by [tex]\(-15^\circ\)[/tex] is the point:
[tex]\[ \boxed{(0,0)} \][/tex]
1. Identify the coordinates of point [tex]\( P \)[/tex] before rotation: Given point [tex]\( P \)[/tex] is at the origin, its coordinates are [tex]\( (0, 0) \)[/tex].
2. Define the rotation angle: The rotation angle is [tex]\(-15^\circ\)[/tex].
3. Construct the rotation matrix: Rotation matrices are used to handle transformations involving rotations about the origin. The general form of the rotation matrix for an angle [tex]\(\theta\)[/tex] is:
[tex]\[ R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \][/tex]
Here, [tex]\(\theta\)[/tex] is [tex]\(-15^\circ\)[/tex].
4. Apply the rotation matrix to point [tex]\( P \)[/tex]: To find the new position of point [tex]\( P \)[/tex], we multiply the rotation matrix by the coordinate vector of [tex]\( P \)[/tex].
5. Compute the new coordinates: By multiplying the rotation matrix by the coordinates [tex]\((0, 0)\)[/tex], we get:
[tex]\[ \begin{pmatrix} \cos(-15^\circ) & -\sin(-15^\circ) \\ \sin(-15^\circ) & \cos(-15^\circ) \end{pmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot \cos(-15^\circ) + 0 \cdot -\sin(-15^\circ) \\ 0 \cdot \sin(-15^\circ) + 0 \cdot \cos(-15^\circ) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \][/tex]
Thus, rotating the point [tex]\((0, 0)\)[/tex] by [tex]\(-15^\circ\)[/tex] does not change its position. The coordinates of the image of point [tex]\( P \)[/tex] remain [tex]\((0, 0)\)[/tex].
So, the image of [tex]\( P \)[/tex] after rotation by [tex]\(-15^\circ\)[/tex] is the point:
[tex]\[ \boxed{(0,0)} \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.