The Student Room Group
Reply 1
Short answer
r' = (a x r) + b for some constant vector b

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Long answer
We know that

r''(1) = a(2)r'(3) - a(3)r'(2)
r''(2) = a(3)r'(1) - a(1)r'(3)
r''(3) = a(1)r'(2) - a(2)r'(1)

(where, for example, r'(1) is the first component of r'). Integrating each of these three scalar differential equations,

r'(1) = a(2)r(3) - a(3)r(2) + b(1)
r'(2) = a(3)r(1) - a(1)r(3) + b(2)
r'(3) = a(1)r(2) - a(2)r(1) + b(3)

for some constants b(1), b(2), b(3). Putting the components back together,

r' = (a x r) + b