 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/15/d/solution

Put

D = [a, m, n] \neq = 0

and rewrite the intersection equation as

b = a + s m - t n

a

and

n

[a, b, n] = [a, a + s m - t n, n] = sD .

Taking the triple product with

a

and

m

instead gives

[a, b, m] = [a, a + s m - t n, m] = - t [a, n, m] = t D .

s = \frac{[ a , b , n ]}{[ a , m , n ]}, t = \frac{[ a , b , m ]}{[ a , m , n ]} .

The assumed nonzero denominator makes these expressions valid and guarantees nonparallel directions. The resulting point is

r = a + s m = b + t n

; these formulas are conditional on the intersection assumed in the question.

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