if x,y and z are not all zero and connected by the equations `a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0` and `(p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0` show that
`lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|`

Explanation

Since x,y and z are not all zero the determinant of the coefficient of the given set of equation must satisfy
` |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(p_(1)+lambdaq_(1),,p_(2)+lambdaq_(2),,p_(3)+lambdaq_(3)):}|=0`
` rArr lambda= |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(p_(1),,p_(2),,p_(3)):}|-: |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(q_(1),,q_(2),,q_(3)):}|`