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Position Vector

If a⃗,b⃗,c⃗...

Question

If $\to a, \to b, \to c, \to d, \to e, \to f$ are position vectors of 6 points A, B, C, D, E & F respectively such that $3 \to a + 4 \to b = 6 \to c + \to d = 4 \to e + 3 \to f = \to x$ then

A

¯ ¯¯¯¯¯¯ ¯ A B

is parallel to

¯ ¯¯¯¯¯¯¯ ¯ C D

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B

line AB, CD and EF are concurrent

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C

\frac{\to x}{7}

is position vector of the point dividing CD in ratio 1 : 6

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D

A, B, C, D, E & F are coplanar

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Solution

The correct options are
B line AB, CD and EF are concurrent
C $\frac{\to x}{7}$ is position vector of the point dividing CD in ratio 1 : 6
Given condition
$3 \to a + 4 \to b = 6 \to c + \to d = 4 \to e + 3 \to f = \to x$
It is clearly seen that sum of coefficient
$7 = 7 = 7 = \to x$
So lines are concurrent
position vector of c and d
$p o s i t i o n v e c t o r = \frac{6 c + d}{7}$
from above condition
$p o s i t i o n v e c t o r = \frac{x}{7}$

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