MCQ
If four distinct points with position vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are coplanar; then $[\vec{a} \vec{b} \vec{c}]$ is equal to
  • $[\overrightarrow{ d } \overrightarrow{ c } \overrightarrow{ a }]+[\overrightarrow{ b } \overrightarrow{ d } \overrightarrow{ a }]+[\overrightarrow{ c } \overrightarrow{ d } \overrightarrow{ b }]$
  • B
    $[\overrightarrow{ d } \overrightarrow{ b } \overrightarrow{ d }]+[\overrightarrow{ a } \overrightarrow{ c } \overrightarrow{ d }]+[\overrightarrow{ d } \overrightarrow{ b } \overrightarrow{ c }]$
  • C
    $[\overrightarrow{ a } \overrightarrow{ d } \overrightarrow{ b }]+[\overrightarrow{ d } \overrightarrow{ c } \overrightarrow{ a }]+[\overrightarrow{ d } \overrightarrow{ b } \overrightarrow{ c }]$
  • D
    $[\overrightarrow{ b } \overrightarrow{ c } \overrightarrow{ d }]+[\overrightarrow{ d } \overrightarrow{ a } \overrightarrow{ c }]+[\overrightarrow{ d } \overrightarrow{ b } \overrightarrow{ a }]$

Answer

Correct option: A.
$[\overrightarrow{ d } \overrightarrow{ c } \overrightarrow{ a }]+[\overrightarrow{ b } \overrightarrow{ d } \overrightarrow{ a }]+[\overrightarrow{ c } \overrightarrow{ d } \overrightarrow{ b }]$
a
$\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are coplanar points. $\vec{b}-\vec{a}, \vec{c}-\vec{a}, \vec{d}-\vec{a}$ are coplanar vectors.

So, $[\overrightarrow{ b }-\overrightarrow{ a } \overrightarrow{ c }-\overrightarrow{ a } \overrightarrow{ d }-\overrightarrow{ a }]=0$

$(\overrightarrow{ b }-\overrightarrow{ a }) \cdot((\overrightarrow{ c }-\overrightarrow{ a }) \times(\overrightarrow{ d }-\overrightarrow{ a }))=0$

${[\overrightarrow{ b } \overrightarrow{ c } \overrightarrow{ d }]-[\overrightarrow{ b } \overrightarrow{ c } \overrightarrow{ a }]-[\overrightarrow{ b } \overrightarrow{ a } \overrightarrow{ d }]-[\overrightarrow{ a } \overrightarrow{ c } \overrightarrow{ d }]=0}$

$\Rightarrow[\overrightarrow{ a } \overrightarrow{ b } \overrightarrow{ c }]=[\overrightarrow{ c } \overrightarrow{ d } \overrightarrow{ b }]+[\overrightarrow{ b } \overrightarrow{ d } \overrightarrow{ a }]+[\overrightarrow{ d } \overrightarrow{ c } \overrightarrow{ a }]$

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