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Condition for Coplanarity of Four Points

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Question

Let $\to a, \to b$ and $\to c$ be three non-zero vectors, two of which are collinear. If the vector $3 \to a + 7 \to b$ is collinear with $\to c$ and $3 \to b + 2 \to c$ is collinear with $\to a$ , then $9 \to a + 21 \to b + 14 \to c$ is equal to-

A

λ \to a

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B

λ \to c

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C

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D

none of these

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Solution

The correct option is C 0
$\to a, \to b, \to c$ Now zero vector
$3 \to a + 7 \to b$ is collinear with $\to c$
$3 \to b + 2 \to c$ is collinnear with $\to a$
$9 \to a + 21 \to b + 14 \to c$

$3 \to a + 7 \to b = p \to c$
$3 \to b + 2 \to c = q \to a$
$3 \to b + 2 \to c = q (\frac{p \to c - 7 \to b}{3})$

$9 \to b + 6 \to c = q p \to c - 7 q \to b$

$(9 + 7 q) \to b = \to c (q p - 6)$ is b and c is not collinear so

$q - 9 = 79$
$q = - 9 / 7$
$3 \to b + 2 \to c = q \to a$
$3 \to b + 2 \to c = \frac{- 9}{7} \to a$
$21 \to b + 14 \to c = - 9 \to a$
$9 \to a + 21 \to b + 14 \to c = 0$

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Condition for Coplanarity of Four Points

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