# Point A is at #(1 ,3 )# and point B is at #(-7 ,-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?

We can produce a rotation about the origin by using the transformation matrix:

This is for anticlockwise rotation, so for clockwise rotation we use the angle:

So we have:

Distance between A and B:

Distance between A' and B:

Change in distance:

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To rotate point A, (1, 3), clockwise about the origin by π/2 radians, we 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 ( \theta = \pi/2 ), and ( x = 1 ), ( y = 3 ).

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \cos(\pi/2) & -\sin(\pi/2) \ \sin(\pi/2) & \cos(\pi/2) \end{pmatrix} \begin{pmatrix} 1 \ 3 \end{pmatrix} ]

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \ 3 \end{pmatrix} ]

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 0 \ 1 \end{pmatrix} ]

So, the new coordinates of point A after rotating π/2 radians clockwise about the origin are (0, 1).

To find the change in distance between points A and B, we need to calculate the distance between the original point A and point B, and then between the new point A and point B.

Original distance between A and B: [ d_{AB} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} ] [ d_{AB} = \sqrt{(-7 - 1)^2 + (-5 - 3)^2} ] [ d_{AB} = \sqrt{(-8)^2 + (-8)^2} ] [ d_{AB} = \sqrt{128} ] [ d_{AB} = 8\sqrt{2} ]

New distance between A and B:
[ d'*{AB} = \sqrt{(x_B - x' A)^2 + (y_B - y'A)^2} ]
[ d'{AB} = \sqrt{(-7 - 0)^2 + (-5 - 1)^2} ]
[ d'{AB} = \sqrt{(-7)^2 + (-6)^2} ]
[ d'*{AB} = \sqrt{49 + 36} ]
[ d'_{AB} = \sqrt{85} ]

The change in distance between points A and B is:
[ \Delta d_{AB} = d'*{AB} - d*{AB} ]
[ \Delta d_{AB} = \sqrt{85} - 8\sqrt{2} ]

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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.

- What is a transformation? And what are the four types of transformations?
- Point A is at #(3 ,7 )# and point B is at #(3 ,-4 )#. 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?
- A triangle has corners at #(8 ,3 )#, #(4 ,-5 )#, and #(2 ,1 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-3 ), how far will its centroid move?
- Points A and B are at #(7 ,9 )# and #(6 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- A line segment with endpoints at #(2 , 2 )# and #(5, -3 )# is rotated clockwise by #pi #. What are the new endpoints of the line segment?

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