Rotation Formula

Think of a compass and draw a circle, the point where you put the pin to rotate the compass to draw the circle, is the point which is called as a “centre of rotation”.

The rotation turns the circle through an angle. Rotation can be done clockwise as well as counterclockwise. The most common rotation angles are 90 degrees, 180 degrees, 270 degrees etc.

We have the formula for the three angles of rotation below:

\[\large Rotation\;90^{\circ}: R_{90^{\circ}}(x,y) = (-y,x)\]
\[\large Rotation\;180^{\circ}: R_{180^{\circ}}(x,y) = (-x,-y)\]
\[\large Rotation\;270^{\circ}: R_{270^{\circ}}(x,y) = (y,-x)\]

Solved example

Question: What are the angles formed if the rotation is done at 270^{\circ} clockwise?

i. 12, 4
ii. 15, 8
iii. 3, 2

Through the formulas we can tell that the rotation has been done in angle:
i .  12, 4 = -12, 4
ii. 15, 8 = -15, 8
iii. 3, 2 = -3, 2

Practise This Question

A thin wire of length 'L' and uniform linear mass density ρ is bent into a circular loop with centre at O as shown in the figure.  The moment of inertia of the loop about the axis XX' is