# Point A is at #(5 ,2 )# and point B is at #(2 ,-4 )#. Point A is rotated #(3pi)/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?

New coordinate of

Reduction in distance due to rotation around origin is

Point A (5,2), Point B (2, -4)

Point A rotated by

Point A moves from Quadrant I to Quadrant II

Distance formula

Reduction in distance due to rotation around origin is

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To find the new coordinates of point A after rotating it (3π)/2 clockwise about the origin, you can use the rotation matrix formula:

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} ]

where (x, y) are the original coordinates of point A, (x', y') are the new coordinates after rotation, and θ is the angle of rotation. Given that the angle of rotation is (3π)/2, the cosine and sine of this angle are 0 and -1 respectively. Thus, the rotation matrix becomes:

[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} ]

For point A (5, 2), applying this rotation matrix:

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 5 \ 2 \end{pmatrix} ]

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} -2 \ 5 \end{pmatrix} ]

So, the new coordinates of point A after the rotation are (-2, 5).

To find the change in distance between points A and B, you calculate the distances before and after rotation and then find the difference. The distance formula between two points (x1, y1) and (x2, y2) is:

[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} ]

For points A (5, 2) and B (2, -4):

Before rotation: [ d_{AB} = \sqrt{(2 - 5)^2 + ((-4) - 2)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} ]

After rotation (A' (-2, 5)): [ d_{A'B} = \sqrt{(2 - (-2))^2 + ((-4) - 5)^2} = \sqrt{(2 + 2)^2 + (-9)^2} = \sqrt{16 + 81} = \sqrt{97} ]

The change in distance is: [ \Delta d = d_{A'B} - d_{AB} = \sqrt{97} - \sqrt{45} ]

Therefore, the change in distance between points A and B after rotation is ( \sqrt{97} - \sqrt{45} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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