In this lesson, we’ll discuss the rotation of the coordinate axes about the origin.
That is, how do the coordinates of a point P(x, y) change if the axes are rotated about the origin by an angle θ?
Rotation of Axes
Suppose there’s a point P(x, y) on the XY plane.
And, suppose we rotate the axes origin by an angle θ in the anticlockwise direction.
What will be the coordinates of P with respect to the new axes?
To find the new coordinates, i.e. x’ and y’, we need to the distance of P from the rotated axes, in terms of x, y, and θ.
Let’s do the calculations. Here’s a figure to help.
Let’s zoom in a bit and label some points.
I’ve also dropped perpendiculars from D to OX and PB. Okay, stay with me.
Now, PA = x and PB = y (the old coordinates). Also, PC = x’ and PD = y’ (the new coordinates).
In ΔPFD, we have ∠DPF = θ (figure out why). This means that
PF = y’cosθ …I
Also,
FD = y’sinθ
⇒ BE = y’sinθ (since FD = BE) …II
In ΔDOE,
OE = ODcosθ
⇒ OE = CPcosθ (since OD = CP)
⇒ OE = x’cosθ …III
Also,
DE = ODsinθ
⇒ DE = x’sinθ (since OD = CP = x’)
⇒ FB = x’sinθ (since FB = DE) …IV
We’re almost done. Let’s bring back the figure.
Now,
PA = OB
⇒ x = OB (since PA = x)
⇒ x = OE – BE
⇒ x = x’cosθ – y’sinθ (using II and III)
Similarly,
y = PB
⇒ y = FB + PF
⇒ y = x’sinθ + y’cosθ (using IV and I)
And that’s it! We’ve obtained the relation between the old coordinates (x and y) and the new ones (x’ and y’):
x = x’cosθ – y’sinθ
y = x’sinθ + y’cosθ
You might want to revisit the calculations. I suggest you try deriving this again on your own.
We can also express x’ and y’ explicitly in terms of x and y. I’ll skip the calculations here. We’ll get:
x’ = xcosθ + ysinθ
y’ = ycosθ – xsinθ
Here’s a simulation that shows how the coordinates of a point change when the axes are rotated about the origin.
You can drag the blue point to change the angle of rotation. The point P can also be dragged. Do the old and new coordinates satisfy the relation that we derived?
Lesson Summary
Let P(x, y) be a point on the XY plane. Let the axes be rotated about origin by an angle θ in the anticlockwise direction. Then with respect to the rotated axes, the coordinates of P, i.e. (x’, y’), will be given by:
x = x’cosθ – y’sinθ
y = x’sinθ + y’cosθ
The next lesson will discuss a few examples related to translation and rotation of axes. See you there!