The position vectors of the points A,B,C are respectively a¯,b¯,c¯ . If p devides AB in the ratio 3:4 and Q devides BC in the ratio 2:1 both externally then PQ =

OA¯=a,¯OB¯=b,¯OC¯=c,¯ now OP¯=3b¯−4a¯3−4=4a¯−3b¯ OQ¯=2c¯−b¯2−1=2c¯−b¯ now PQ¯=OQ¯−OP¯ =2c¯+2b¯−4a¯

The position vectors of the points A,B,C are respectively a¯,b¯,c¯ . If p devides AB in the ratio 3:4 and Q devides BC in the ratio 2:1 both externally then PQ =

OA¯=a,¯OB¯=b,¯OC¯=c,¯ now OP¯=3b¯−4a¯3−4=4a¯−3b¯ OQ¯=2c¯−b¯2−1=2c¯−b¯ now PQ¯=OQ¯−OP¯ =2c¯+2b¯−4a¯

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The position vectors of the points A,B,C are respectively $\bar{a}, \bar{b}, \bar{c}$ . If p devides $AB$ in the ratio 3:4 and Q devides $BC$ in the ratio 2:1 both externally then $PQ$ =

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$b + c - 2 a$

2( $b + c - 2 a$ )

$4 a - b - c$

$\frac{- 2 \bar{a} - \bar{b} - \bar{c}}{2}$

answer is B.

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Detailed Solution

$\bar{OA} = \bar{a,} \bar{OB} = \bar{b,} \bar{OC} = \bar{c,} n o w \bar{OP} = \frac{3 \bar{b} - 4 \bar{a}}{3 - 4} = 4 \bar{a} - 3 \bar{b} \bar{OQ} = \frac{2 \bar{c} - \bar{b}}{2 - 1} = 2 \bar{c} - \bar{b} n o w \bar{PQ} = \bar{OQ} - \bar{OP} = 2 \bar{c} + 2 \bar{b} - 4 \bar{a}$