15. Rotation of Axes

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.

rotation of axes

And, suppose we rotate the axes origin by an angle θ in the anticlockwise direction.

rotation of axes

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.

rotation of axes

Let’s zoom in a bit and label some points.

rotation of axes

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.

rotation of axes

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!

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