Prove that vectors
`vecu=(al+a_(1)l_(1))hati+(am+a_(1)m_(1))hatj + (an+a_(1)n_(1))hatk`
`vecv=(bl+b_(1)l_(1))hati+(bm + b_(1)m_(1))hatj+(bn+b_(1)n_(1))hatk`
`vecw=(wl+c_(1)l_(1))hati+(cm+c_(1)m_(1))hatj+(cn+c_(1)n_(1))hatk`

Explanation

`[vecuvecv vecw]=|{:(al=a_(1)l_(1),am+a_(1)m_(1),an+a_(1)n_(1)),(bl+b_(1)l_(1),bn+b_(1)m_(1),bn+b_(1)n_(1)),(cl+c_(1)l_(1),cm+c_(1)m_(1),cn+c_(1)n_(1)):}|`
`=|{:(a,a_(1),0),(b,b_(1),0),(c,c_(1),0):}||{:(l,l_(1),0),(m,m_(1),0),(n,n_(1),0):}|=0`
Therefore, the given vectors are coplanar.