If A-B-C-D are any four points in space- then -AB-CD - BC-AD -CA-BD- m-Area of ABC- Find m-

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Standard XII

Mathematics

Multiplication of a Vector by a Scalar

If A,B,C,D ...

Question

If $A, B, C, D$ are any four points in space, then $∣ ∣ ∣ - - \to A B \times - - \to C D + - - \to B C \times - - \to A D + - - \to C A \times - - \to B D ∣ ∣ ∣ =$ $m \times$ Area of $△ A B C$ . Find m.

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Solution

$\to a, \to b, \to c, \to d$ be the position vectors of $A, B, C, D$
$△ A B C = \frac{1}{2} ∣ ∣ ∣ (- - \to A B \times - - \to A C) ∣ ∣ ∣$
$= \frac{1}{2} ∣ ∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣ ∣$
Again $∣ ∣ ∣ - - \to A B \times - - \to C D + - - \to B C \times - - \to A D + - - \to C A \times - - \to B D ∣ ∣ ∣$
$= ∣ ∣ ∣ (\to b - \to a) \times (\to d - \to c) + (\to c - \to b) \times (\to d - \to a) + (\to a - \to c) \times (\to d - \to b) ∣ ∣ ∣$
$= 2 ∣ ∣ ∣ \to b \times \to c + \to c \times \to a + \to a \times \to b ∣ ∣ ∣$
$= 4 \times$ Area of $△ A B C$

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If A,B,C,D are any four points in space, then ∣∣∣−−→AB×−−→CD+−−→BC×−−→AD+−−→CA×−−→BD∣∣∣=m×Area of △ABC. Find m.

If $A, B, C, D$ are any four points in space, then $∣ ∣ ∣ - - \to A B \times - - \to C D + - - \to B C \times - - \to A D + - - \to C A \times - - \to B D ∣ ∣ ∣ =$ $m \times$ Area of $△ A B C$ . Find m.