Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / iv

Table of contents
- Solution iv

Solution

Put

L = ∣ x_{n} ∣

and choose perpendicular unit vectors

e, f

such that

x_{n} = L e

and

x_{n + 1} = λ L f

. Direction vectors for the two lines are

x_{n + 2} - x_{n + 1} = - λ L (λ e + f)

and

x_{n + 3} - x_{n} = - L (e + λ^{3} f) .

Their scalar product and norms give

cos θ = \frac{λ ^{2} L ^{2} ( 1 + λ ^{2} )}{λ L ^{2} 1 + λ ^{2} 1 + λ ^{6}} .

Therefore the acute angle between the straight lines satisfies

cos θ = \frac{λ 1 + λ ^{2}}{1 + λ ^{6}} .

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