If A - B - C - D are four points in a space and - AB - CD - BC - AD - CA - BD - area of the triangle ABC - Then the value of - isA- 1B- 2C- 3D- 4

The correct option is D 4 Let D be origin and position vectors of A, B, and C be a, b and c respectively. ⇒ |AB×CD+BC×AD+CA×BD| =|(b a)×( c)+(c b)×( a)+(a c)× ...

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Byju's Answer

Standard XII

Mathematics

Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

If A , B , C ...

Question

If $A, B, C, D$ are four points in a space and $| A B \times C D + B C \times A D + C A \times B D | = λ$ (area of the triangle ABC). Then the value of $λ$ is

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Solution

The correct option is D
4

Let $D$ be origin and position vectors of $A, B,$ and $C$ be $\to a, \to b$ and $\to c$ respectively.

$\Rightarrow | - - \to A B \times - - \to C D + - - \to B C \times - - \to A D + - - \to C A \times - - \to B D |$

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

$= 2 | \to a \times \to b + \to b \times \to c + \to c \times \to a |$

$= 2 \times 2$ area of $Δ$ ABC

Hence, $λ = 4$

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If A, B, C, D are four points in a space and |AB×CD+BC×AD+CA×BD|=λ(area of the triangle ABC). Then the value of λ is

If $A, B, C, D$ are four points in a space and $| A B \times C D + B C \times A D + C A \times B D | = λ$ (area of the triangle ABC). Then the value of $λ$ is