Demonstrating the properties of rotations, if a line segment with endpoints (0, - 3) and (0, - 7) is rotated 90 deg clockwise, what is an endpoint of this rotated segment? (point)

Answer 1

Let's start by visualizing the original line segment with endpoints (0, -3) and (0,-7).

The line segment is vertical and goes down from coordinate (0, -3) to (0,-7).

When we rotate this line segment 90 degrees clockwise, we need to find the new coordinates of the endpoint.

To do this, we need to apply the rotation matrix:

[ cosθ -sinθ ]
[ sinθ cosθ ]

In this case, since we are rotating 90 degrees clockwise, we have:

[ cos(90) -sin(90) ]
[ sin(90) cos(90) ]

Now let's substitute the rotation matrix with the values:

[ 0 -1 ]
[ 1 0 ]

Applying the rotation matrix to the endpoint (0,-7):
[ 0 -1 ][0] = [ 0 * 0 + (-1) * (-7) ] = [ 0 + 7 ] = [ 7 ]

So, after rotating the line segment 90 degrees clockwise, the new endpoint is (0, 7).