The length of the perpendicular from the origin to the plane passing through three non-collinear points with position vector $\to a, \to b, \to c$ is

\frac{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}{∣ ∣ ∣ \to a \times \to b + \to c \times \to a + \to b \times \to c ∣ ∣ ∣}

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\frac{2 [\begin{matrix} \to a & \to b & \to c \end{matrix}]}{∣ ∣ ∣ \to a \times \to b + \to c \times \to a + \to b \times \to c ∣ ∣ ∣}

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[\begin{matrix} \to a & \to b & \to c \end{matrix}]

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None of these

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Solution

The correct option is A $\frac{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}{∣ ∣ ∣ \to a \times \to b + \to c \times \to a + \to b \times \to c ∣ ∣ ∣}$
The vector equation of the plane passing through points with position vectors $\to a, \to b$ and $\to c$ is
$\to r . (\to a \times \to b + \to c \times \to a + \to b \times \to c) = [\begin{matrix} \to a & \to b & \to c \end{matrix}]$
Therefore, the length of the perpendicular from the origin to this plane is given by

$\frac{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}{∣ ∣ ∣ \to a \times \to b + \to c \times \to a + \to b \times \to c ∣ ∣ ∣}$

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