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?

Correct answer: B. 1:3 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

NDA 2025

Exam: NDA
Year: 2025
Subject: Mathematics
Chapter: Vector Algebra

NDA 2025 · Mathematics · Vector Algebra

The position vectors of three points A , B and C are a , b and c...

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?

Options

A $3:1$
B $1:3$
C $3:4$
D $1:4$

Correct answer

B. $1:3$

Step-by-step solution

Given

3\vec{a} - 4\vec{b} + \vec{c} = \vec{0}

Rearranging the terms:

3\vec{a} - 3\vec{b} = \vec{b} - \vec{c}

-3(\vec{b} - \vec{a}) = -(\vec{c} - \vec{b})

Using the definition of position vectors,

\overrightarrow{AB} = \vec{b} - \vec{a}

and

\overrightarrow{BC} = \vec{c} - \vec{b}

-3\overrightarrow{AB} = -\overrightarrow{BC}

\overrightarrow{BC} = 3\overrightarrow{AB}

Taking the magnitude on both sides:

|\overrightarrow{BC}| = 3|\overrightarrow{AB}|

BC = 3AB

\dfrac{AB}{BC} = \dfrac{1}{3}

Thus,

AB : BC = 1 : 3

.

Answer:

1:3

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