Rotation of Axes


This lesson will discuss rotation of the coordinate axes about the origin. Something like this:

Rotation of Axes Coordinate Geometry

Rotation of Axes

Consider a point P(x, y), and let’s suppose that the axes have been rotated about origin by an angle θ in the anticlockwise direction. What will be the coordinates of the point P, with respect to the new axes?

To find the new coordinates, we’ll find the distances of the point from the rotated axes. Let’s do the calculations. Let (x, y) be the original coordinates and (X, Y) be the final ones. The following figure well help us.

Rotation of Axes Coordinate Geometry

It’ll get a little messy. Let’s zoom in a bit. I’ve labeled the points nicely – have a close look.

Rotation of Axes Coordinate Geometry

Told you it’ll get messy. Let’s begin with the calculations: PA = x and PB = y (the old coordinates). Next, PC = X, PD = Y (the new coordinates). I’ve dropped perpendiculars from D to the old X axis, and to PB (DE and DF respectively).

Now PF = Ycosθ, and FD = BE = Ysinθ. Lastly, OE = ODcosθ = CPcosθ = Xcosθ. And, FB = DE = ODsinθ = Xsinθ.

And we’re almost done.

Now AP = x = OB, which can be written as OE – BE, that is Xcosθ – Ysinθ. Therefore, x = Xcosθ – Ysinθ.

Next, OA = y = PB, which can be written as FB + PF = Xsinθ + Ycosθ. In other words, y = Xsinθ + Ycosθ.

And that’s it! We’ve obtained the relation between the old coordinates (x and y) and the new ones (X and Y).

Lesson Summary

Let P(x, y) be a point in the X-Y plane. If the axes are rotated by an angle θ in the anticlockwise direction about the origin, then the coordinates of P with respect to the rotated axes will be given by the following relations:

x = Xcosθ – Ysinθ

y = Xsinθ + Ycosθ.

where (X, Y) denote the new coordinates of P.

The next lesson will discuss a few examples related to translation and rotation of axes. See you there!

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