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Condition for Coplanarity of Four Points

Let a,b,c b...

Question

Let $\to a, \to b, \to c$ be three non-coplanar vectors and $\to r$ be any vector in space such that $\to r . \to a = 1, \to r . \to b = 2$ and $\to r . \to c = 3.$ If $[\to a, \to b, \to c] = 1$ then $\to r$ is equal to

A

\to a + 2 \to b + 3 \to c

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B

\to b \times \to c + 2 \to c \times \to a + 3 \to a \times \to b

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C

(\to b . \to c) \to a + 2 (\to c . \to a) \to b + 3 (\to a . \to b) . \to c

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D

none of the above

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Solution

The correct option is C $\to b \times \to c + 2 \to c \times \to a + 3 \to a \times \to b$
Let $\to r = l (\to b \times \to c) + m (\to c \times \to a) + n (\to a \times \to b)$
$\to r . \to a = l (\to b \times \to c) \to a + m (\to c \times \to a) \to a + n (\to a \times \to b) \to a$
$\to r . \to a = l (\to b \times \to c) \to a + 0 + 0$ since cross product of like vectors $= 0$
$\Rightarrow 1 = l [\to a, \to b, \to c]$
$\Rightarrow l = 1$
$\to r . \to b = l (\to b \times \to c) \to b + m (\to c \times \to a) \to b + n (\to a \times \to b) \to b$
$\to r . \to b = 0 + m (\to c \times \to a) \to b + 0$ since cross product of like vectors $= 0$
$\Rightarrow 2 = m [\to a, \to b, \to c]$
$\Rightarrow m = 2$
$\to r . \to c = l (\to b \times \to c) \to c + m (\to c \times \to a) \to c + n (\to a \times \to b) \to c$
$\to r . \to c = 0 + 0 + n (\to a \times \to b) \to c$ since cross product of like vectors $= 0$
$\Rightarrow 3 = n [\to a, \to b, \to c]$
$\Rightarrow n = 3$
$∴ \to r = \to b \times \to c + 2 \to c \times \to a + 3 \to a \times \to b$ where $l = 1, m = 2$ and $n = 3$

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