Rotation: a 90 counterclockwise rotation about the origin maps (x,y)to(,)?

[17593]

what would a 90 counterclockwise rotation about the origin maps (x,y)to(,)?

 

[ksdhart2]

A good start might be to deal with one specific point before generalizing to variables. So, let's pick an arbitrary point P... how about (3,5)? Which quadrant of the plane is this point in? As a hint, consider what you know about the quadrants. Specifically, I'd note that both the x- and y-coordinates are positive. Now, after a 90-degree clockwise rotation about the origin, what quadrant would the new point be in? What do you know about the x- and y-coordinates of points in this quadrant? What does that suggest the coordinates of the new point (call it P' or P prime)? Can you see how to apply this information to a generic point (x,y)? If not, perhaps consider a starting point in quadrant 2 and repeat the above steps. If you need a refresher on the quadrants, try here.

 

[pka]

what would a 90 counterclockwise rotation about the origin maps (x,y)to(,)?

Transform point \(\displaystyle P: (x,y) \) through a \(\displaystyle \frac{\pi}{2} \) counter-clockwise to point \(\displaystyle Q: (?,?) \).
The lines \(\displaystyle \overleftrightarrow {OP}~\&~\overleftrightarrow {OQ} \) through the origin \(\displaystyle O \) are perpendicular to each other.
The line \(\displaystyle \overleftrightarrow {OP}\) has the slope \(\displaystyle \frac{y}{x} \), thus what is the slope of \(\displaystyle \overleftrightarrow {OQ}~? \)

 

[Ishuda]

what would a 90 counterclockwise rotation about the origin maps (x,y)to(,)?

Another way: To give the transformation some co-ordinates, say we have
(x,y) -----> (u,v)
via a counter-clockwise rotation about (x,y)=(0,0). The y axis swings into the -u axis [-u=y or u=-y] and the x axis swings into the v [v=x] axis so
(u,v) = (-y, x)

Another way [although I'm not sure you have had this yet]: A simple rotation about (x,y)=0 is given by
(x,y) -----> (\(\displaystyle cos(\theta)\) x - \(\displaystyle sin(\theta)\) y, \(\displaystyle sin(\theta)\) x + \(\displaystyle cos(\theta) y\))
where \(\displaystyle \theta\) is measured from the x axis and counter-clockwise is positive.

 

[Bob Brown MSEE]

Try some simple examples.

what would a 90 counterclockwise rotation about the origin maps (x,y)to(,)?

4 simple examples of 90 degree counterclockwise rotation about the origin
1) maps ( 1, 0) to ( 0, 1)
2) maps ( 0, 1) to (-1, 0)
3) maps (-1, 0) to ( 0,-1)
4) maps ( 0,-1) to ( 1, 0)

Let map variables be called (x,y) to (x',y')

Assume that the map is linear, then...
x' = ax + by
y' = cx + dy

From example 1)
0 = a1 + b0
1 = c1 + d0
note: a=0 and c=1

From example 2)
-1 = a0 + b1
0 = c0 + d1
note: b=-1 and d=0

So now we know a,b,c and d

RULE: (x,y) to (-y,x)
x' = - y
y' = x
note: this RULE agrees with examples 3) and 4)
try some other values for (x,y) and see if they work!