What is the image of the point (4,-6) after a rotation of 90 counterclockwise about the origin
(x,y)→(y,-x)
To find the image of a point after a rotation of 90 degrees counterclockwise about the origin, we can use the following steps:
1. Plot the point (4, -6) on a coordinate plane.
2. Draw a line connecting the origin (0, 0) and the point (4, -6).
3. Since we want to rotate counterclockwise, we can visualize the point moving in a circular path around the origin.
4. A rotation of 90 degrees counterclockwise means that the point will end up on the y-axis, but with an opposite sign for the x-coordinate.
5. The final image point will have an x-coordinate of 6 and a y-coordinate of 4.
6. Therefore, the image of the point (4, -6) after a rotation of 90 degrees counterclockwise about the origin is (6, 4).
Note: Remember to label the point on the coordinate plane after the rotation to make it clear what the image point is.
To find the image of a point after a rotation, you can perform the following steps:
1. Identify the coordinates of the initial point.
2. Apply the rotation transformation to the point using a rotation matrix.
3. Determine the coordinates of the new point, which represents the image after the rotation.
In this case, the initial point is (4, -6), and we want to rotate it 90 degrees counterclockwise about the origin. To perform this rotation, we can use a rotation matrix.
The rotation matrix for a counterclockwise rotation of angle θ about the origin is given by:
| cosθ -sinθ |
| sinθ cosθ |
For a rotation of 90 degrees counterclockwise, we have θ = 90°.
Applying the rotation matrix to the coordinates (4, -6), we have:
| cos90° -sin90° | | 4 |
| sin90° cos90° | * |-6 |
Calculating the values of cos90°, sin90°, -sin90°, and cos90°, we get:
| 0 -1 | | 4 |
| 1 0 | * |-6 |
Multiplying the matrices, we have:
| 0 * 4 + -1 * -6 |
| 1 * 4 + 0 * -6 |
Simplifying, we obtain:
| 6 |
| 4 |
Therefore, the image of the point (4, -6) after a rotation of 90 degrees counterclockwise about the origin is (6, 4).