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?

Question

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?

Aurora Ayala · Accepted Answer

See below.

It can be seen from the diagram, that a rotation about the origin through an angle #theta# can be represented as:

#((1),(0))->((costheta),(sintheta))# and #((0),(1))->((-sintheta),(color(white)(8)costheta))#

So the transformation matrix will be:

#((costheta,-sintheta),(sintheta,color(white)(8)costheta))#

Matrix for point A:

#A= ((5),(-2))#

Transformation:- Rotation through #pi/2# clockwise. This is equivalent to #2pi-pi/2=(3pi)/2# anticlocwise.

#((cos((3pi)/2),-sin((3pi)/2)),(sin((3pi)/2),color(white)(8)cos((3pi)/2))) ((color(white)(88)5),(-2))=((-2),(color(white)()-5))#

#A'=((-2),(color(white)()-5))#

Distance between A and B:

#d=sqrt((5-(-2))^2+(-2-(-5))^2)=7sqrt2#

Distance between A' and B:

#d=sqrt((-2-(-2))^2+(-5-5)^2)=10#

The distance between the points has increased by a factor of #(5sqrt(2))/7#

Sophia Alfred · Answer

undefined 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:
$$ \Delta d = |d_2 - d_1| $$