If `veca, vecb and vecc` are any three non-coplanar vectors, then prove that points `l_(1)veca+ m_(1)vecb+ n_(1)vecc, l_(2)veca+m_(2)vecb+n_(2)vecc, l_(3)veca+m_(3)vecb+ n_(3)vecc, l_(4)veca + m_(4)vecb+ n_(4)vecc` are coplanar if `|{:(l_(1),, l_(2),,l_(3),,l_(4)),(m_(1),,m_(2),,m_(3),,m_(4)), (n_1,,n_2,, n_3,,n_4),(1,,1,,1,,1):}|=0`

Explanation

Points `A(l_1veca+m_1vecb+n_1vecc), B(l_2veca+m_2vecb+n_2vecc), C(l_3veca+m_3vecb+n_3vec), D(l_4veca+m_4vecb+n_4vecc)` are coplanar.
`rArr" "` Vectors
`vec(AB)=(l_1-l_2)veca+ (m_1-m_2)vecb+ (n_1-n_2)vecc`,
`vec(AC)= (l_1-l_2)veca+ (m_1-m_3)vecb+ (n_1-n_3)vecc`,
`vec(AD)= (l_1-l_4)veca+ (m_1-m_4)vecb+ (n_1-n_4)vecc`
are coplanar
`rArr" "|{:(l_1-l_2,,m_1-m_2,, n_1-n_2),(l_1-l_3,,m_1-m_3,,n_1-n_3), (l_1-l_3,,m_1-m_3,,n_1-n_4):}|=0`
Now if `|{:(l_1,,l_2,,l_3,,l_4), (m_1,,m_2,,m_3,,m_4), (n_1,,n_2,,n_3,,n_4),(1,,1,,1,,1):}|=0`
Then applying `C_2 to C_2-C_1, C_3 to C_3-C_1, C_4 to C_4 - C_1`, we have
`|{:(l_1,,l_2-l_1,,l_3-l_1,,l_4-l_1), (m_1,,m_2-m_1,,m_3-m_1,,m_4-m_1), (n_1,,n_2-n_1,,n_3-n_1,,n_4-n_1),(1,,0,,0,,0):}|=0`
or `|{:(l_1-l_2,,m_1-m_2,,n_1-n_2), (l_1-l_3,,m_1-m_3,,n_1-n_3), (l_1-l_3,,m_1-m_3,,n_1-n_4):}|=0`