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Perpendicular Distance of a Point from a Plane

The length of...

Question

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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Similar questions

\to a, \to b, \to c

\to a =^i +^j +^k

\to b = - 3^i +^k

\to c =^i + 2^j + 3^k

A)

[\to a \times \to b

\to b \times \to c

\to c \times \to a]

B)

[\to a + \to b

\to b + \to c

\to c + \to a]

C)

[\to a

\to b

\to c]

[\to a

\to b

\to c]^{- 1}

D)

[\to a - \to b

\to b - \to c

\to c - \to a]

[\to a + 2 \to b - \to c, \to a - \to b, \to a - \to b - \to c] =

\to A = \to b \times \to c, \to B = \to c \times \to a, \to C = \to a \times \to b

\to A \times (\to B \times \to C)

\to B \times (\to C \times \to A)

\to C \times (\to A \times \to B)

(\to a + \to b) \times (\to a - \to b) + (\to b + \to c) \times (\to b - \to c) + (\to c + \to a) \times (\to c - \to a) =

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