Given a point (-6,4) in a line segment, what is the new point when the line segment is rotated counterclockwise about the origin 180 degrees
To rotate a point counterclockwise about the origin by 180 degrees, we can multiply the coordinates of the point by the matrix:
\[\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}\]
When we apply this transformation to the point (-6,4), we get:
\[\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix} \begin{bmatrix}
-6 \\
4
\end{bmatrix} = \begin{bmatrix}
-1(-6) + 0(4) \\
0(-6) + (-1)(4)
\end{bmatrix} = \begin{bmatrix}
6 \\
-4
\end{bmatrix}\]
Therefore, the new point after rotating (-6,4) counterclockwise about the origin by 180 degrees is (6,-4).
To find the new point when a point is rotated counterclockwise about the origin, you can use the rotation matrix. The rotation matrix formula for counterclockwise rotation by an angle θ is:
[x', y'] = [x * cos(θ) - y * sin(θ), x * sin(θ) + y * cos(θ)]
In this case, the angle of rotation is 180 degrees, so θ = π. Let's substitute the values:
x = -6
y = 4
θ = π
[x', y'] = [-6 * cos(π) - 4 * sin(π), -6 * sin(π) + 4 * cos(π)]
Now, let's calculate the new values:
[x', y'] = [-6 * (-1) - 4 * 0, -6 * 0 + 4 * (-1)]
Simplifying further:
[x', y'] = [6, -4]
Therefore, when the point (-6, 4) is rotated counterclockwise about the origin 180 degrees, the new point is (6, -4).