The vectors $a = i + j + m k, b = i + j + (m + 1) k$ and $c = i - j + m k$ are coplanar if $m$ is equal to

1

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4

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3

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none of these

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Solution

The correct option is C none of these
$a = i + j + m k$
$b = i + j + (m + 1) k$
$c = i - j + m k$
$a, b, c$ are co-planar.
$\Rightarrow$ det of co-eff matrix $= 0$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & m 1 & 1 & m - 1 1 & - 1 & m \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow (m + m - 1) + 1 + m (- 2) = 0$
$\Rightarrow 2 m - 1 + 1 - 2 m = 0$
Hence the answer is none of these.

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