NDA 1 2025 Maths, question 68

Topic: Vectors

Q68

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

(a) $3:1$
(b) $1:3$ Correct answer
(c) $3:4$
(d) $1:4$

Solution.

Answer: (b) $1:3$

Rearranging gives $\overrightarrow{c} - \overrightarrow{b} = 3\left( \overrightarrow{b} - \overrightarrow{a} \right)$ , or $\overrightarrow{BC} = 3\overrightarrow{AB}$ .
Therefore the lengths satisfy $AB:BC = 1:3$ .