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

Options

  1. A3:1
  2. B1:3
  3. C3:4
  4. D1:4

Correct answer

B. 1:3

Step-by-step solution

Given 3 a - 4 b + c = 0 Rearranging the terms: 3 a - 3 b = b - c -3( b - a ) = -( c - b ) Using the definition of position vectors, AB = b - a and BC = c - b : -3 AB = - BC BC = 3 AB Taking the magnitude on both sides: | BC | = 3| AB | BC = 3AB AB BC = 1 3 Thus, AB : BC = 1 : 3 . Answer: 1:3

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