Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 9 / b / Solution

Let the semicircle have radius $R$, and write $A=|\mathbf{a}|$, $B=|\mathbf{b}|$. The right triangle has hypotenuse $2R$, so the Pythagorean theorem gives $A^2+B^2=4R^2$. Its area is $AB/2$, whereas half the area of the semicircle is $\pi R^2/4$. Equating these and eliminating $R$ givesEquivalently, the relation between the two norms can be writtenThe two possible ratios are reciprocals, corresponding to interchanging the two sides.