demonstrating the properties of rotation if a line segment with endpoint zero 3 and zero and 7 is rotated 90 clockwise what is the endpoint of the rotation segment
To find the endpoint of the line segment after rotating it 90 degrees clockwise, we can use the rotation matrix formula:
[x', y'] = [cos(θ), sin(θ); -sin(θ), cos(θ)] * [x, y]
Given that the initial endpoint is (0, 7) and we want to rotate it 90 degrees clockwise, we have:
θ = -90 degrees = -π/2 radians
Plugging the values into the formula:
[x', y'] = [cos(-π/2), sin(-π/2); -sin(-π/2), cos(-π/2)] * [0, 7]
Using the values of the cosine and sine of -π/2:
cos(-π/2) = 0
sin(-π/2) = -1
Now calculating the matrix multiplication:
[x', y'] = [0, -1; 1, 0] * [0, 7]
This gives us:
x' = 0*0 + (-1)*7 = -7
y' = 1*0 + 0*7 = 0
Therefore, the endpoint of the rotated line segment is (-7, 0).