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

\frac{∣ ∣ ∣ [\to a, \to b, \to c] ∣ ∣ ∣}{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}

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\frac{2 ∣ ∣ ∣ [\to a, \to b, \to c] ∣ ∣ ∣}{∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}

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∣ ∣ ∣ [\to a, \to b, \to c] ∣ ∣ ∣

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\frac{∣ ∣ ∣ [\to a, \to b, \to c] ∣ ∣ ∣}{2 ∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣}

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Solution

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

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