If you multiply the 1x3 vector by all scalars from -infinity to infinity you get all the points on a line. If you do the same for another 1x3 vector, and it is

If you multiply the 1x3 vector by all scalars from -infinity to infinity you get all the points on a line. If you do the same for another 1x3 vector, and it is not parallel to the first, you get all the points on a different line. These two lines define a plane (and the cross product of the two vectors defines its normal vector)