# Point A is at #(5 ,-2 )# and point B is at #(-2 ,5 )#. Point A is rotated #pi/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

See below.

It can be seen from the diagram, that a rotation about the origin through an angle

So the transformation matrix will be:

Matrix for point A:

Transformation:- Rotation through

Distance between A and B:

Distance between A' and B:

The distance between the points has increased by a factor of

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To rotate point A (5, -2) by ( \frac{\pi}{2} ) radians clockwise about the origin, we use the following rotation formula:

[ x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) ] [ y' = x \cdot \sin(\theta) + y \cdot \cos(\theta) ]

Where: ( x' ) and ( y' ) are the new coordinates after rotation, ( x ) and ( y ) are the original coordinates of point A (5, -2), and ( \theta ) is the angle of rotation (in this case, ( \frac{\pi}{2} ) radians clockwise).

Plugging in the values:

[ x' = 5 \cdot \cos\left(\frac{\pi}{2}\right) - (-2) \cdot \sin\left(\frac{\pi}{2}\right) ] [ y' = 5 \cdot \sin\left(\frac{\pi}{2}\right) + (-2) \cdot \cos\left(\frac{\pi}{2}\right) ]

Solving these equations:

[ x' = -2 ] [ y' = 5 ]

So, the new coordinates of point A after rotating ( \frac{\pi}{2} ) radians clockwise about the origin are (-2, 5).

To find the change in distance between points A and B, we need to calculate the distance between the original point A (5, -2) and point B (-2, 5), and then calculate the distance between the new point A (-2, 5) and point B (-2, 5). Finally, we find the difference between these two distances.

Original distance between A and B: [ d_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

New distance between A and B: [ d_2 = \sqrt{(x_2' - x_1)^2 + (y_2' - y_1)^2} ]

Change in distance: [ \Delta d = |d_2 - d_1| ]

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