Given a point (1,2) on a geometric figure, what is the new point when tje figure is rotated clockwise about the origin 180 degrees
To rotate a point (x, y) clockwise about the origin by 180 degrees, the new coordinates can be found using the following formulas:
x' = x * cos(180) - y * sin(180)
y' = x * sin(180) + y * cos(180)
In this case, the point (1, 2) is going to be rotated clockwise by 180 degrees. Substituting these values into the formulas, we get:
x' = 1 * cos(180) - 2 * sin(180)
y' = 1 * sin(180) + 2 * cos(180)
Now, let's calculate the final coordinates:
x' = 1 * (-1) - 2 * 0
= -1
y' = 1 * 0 + 2 * (-1)
= -2
Hence, the new point after rotating (1, 2) clockwise about the origin by 180 degrees is (-1, -2).
To rotate a point about the origin by a certain angle, we can use the rotation matrix:
\[\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\]
In this case, we want to rotate the point (1, 2) clockwise about the origin by 180 degrees. The angle is given in radians by π (since 180 degrees = π radians).
Using the rotation matrix, the new coordinates can be found as:
\[\begin{bmatrix} \cos(π) & -\sin(π) \\ \sin(π) & \cos(π) \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix}\]
Calculating this, we have:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} -1 \cdot 1 + 0 \cdot 2 \\ 0 \cdot 1 + (-1) \cdot 2 \end{bmatrix} = \begin{bmatrix} -1 \\ -2 \end{bmatrix}\]
Therefore, the new point after the rotation is (-1, -2).