Question

Show that the four points $a,b,c,d$ are co-planar, if $\left[\overline{b}\overline{c}\overline{d}\right]+\left[\overline{c}\overline{a}\overline{d}\right]+\left[\overline{a}\overline{b}\overline{d}\right]=\left[\overline{a}\overline{b}\overline{c}\right]$.

Answer 1

Let A, B, C and D be the four points where position vectors are a, b, c, and d respectively.

we have $\overrightarrow{AB}=$ position vector of B-position vector of A

$\overrightarrow{AB}=b-a$

$\overrightarrow{AC}=c-a$

and $\overrightarrow{AD}=d-a$

Now the scaler triple product of the vectors $\overrightarrow{AB},\overrightarrow{AC}$ and $\overrightarrow{AD}$

$=\left[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\right]=$

$=\overrightarrow{AB}.(\overrightarrow{AC}\times \overrightarrow{AD})$

$=(b-a).\left\{\right((c-a)\times (d-a)\left)\right\}$

$=(b-a).(c\times d-c\times a-a\times d+a\times a)$

$=(b-a).(c\times d-c\times a-a\times d)$

$=b.(c\times d)-b.(c\times a)-b.(a\times d)-a.(c\times d)+a.(c\times a)+a.(a\times d)$

$=\left[bcd\right]-\left[bca\right]-\left[bad\right]-\left[acd\right]$

$=\left[bcd\right]+\left[cad\right]+\left[abd\right]-\left[abc\right]$

Now the points A, B, C and D are coplanar if and only of vectors $\overrightarrow{AB},\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar

$\Rightarrow \left[\overrightarrow{AB}\overrightarrow{AC}\overrightarrow{AD}\right]=0$

$\Rightarrow \left[bcd\right]+\left[cad\right]+\left[abd\right]-\left[abc\right]=0$

$\left[bcd\right]+\left[cad\right]+\left[abd\right]=\left[abc\right]$