The position vectors of three points A, B and C are \vec{a}, \vec{b} and \vec{c} respectively such that 3\vec{a}-4\vec{b}+\vec{c}=\vec{0}. What is AB: BC equal to?

Explanation

Rearranging the given equation gives 4\vec{b} = \vec{c} + 3\vec{a} \implies \vec{b} = \frac{1\vec{c} + 3\vec{a}}{1+3}. This means that point B divides the line segment joining A and C internally in the ratio 1:3 (by the section formula). Therefore, AB:BC = 1:3.

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