Given a point on a geometric figure ( 1, 2 ) what is the new point when the point is rotate clockwise about the origin to 90 degrees?
To rotate a point clockwise about the origin by 90 degrees, you can use the following rotation matrix:
| cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
θ is the angle of rotation, in this case 90 degrees.
So, substituting the values:
| cos(90) -sin(90) |
| sin(90) cos(90) |
Simplifying,
| 0 -1 |
| 1 0 |
To apply the rotation matrix to the given point (1, 2), you can multiply the matrix by the coordinates as a column vector:
| 0 -1 | | 1 | | x' |
| 1 0 | * | 2 | = | y' |
Simplifying,
| 0 -1 | | 1 | | -2 |
| 1 0 | * | 2 | = | 1 |
The new point after rotating clockwise by 90 degrees is (-2, 1).
To rotate a point clockwise about the origin by 90 degrees, we can use a rotation matrix. The general formula for a 2D rotation matrix is:
[ cosθ -sinθ ]
[ sinθ cosθ ]
In this case, since we want to rotate the point by 90 degrees clockwise, the angle (θ) would be -90 degrees or -π/2 radians.
Now, let's substitute the values and calculate the new coordinates:
x' = x * cosθ - y * sinθ
= 1 * cos(-π/2) - 2 * sin(-π/2)
= 1 * 0 - 2 * (-1)
= 2
y' = x * sinθ + y * cosθ
= 1 * sin(-π/2) + 2 * cos(-π/2)
= 1 * (-1) + 2 * 0
= -1
Therefore, the new point after rotating the point (1, 2) clockwise by 90 degrees is (2, -1).