) Determine whether the following two planes x + 4y − z = 7 and 5x − 3y −7z = 11 are parallel, orthogonal, coincident (that is, the same) or none of these.

1 answer

Their normals are
(1,4,-1) and (5,-3,-7) respectively

clearly they are not parallel, or one of the normals would be a multiple of the other
nor are they coincident

let's look at their dot product:
(1,4,-1) . (5,-3,-7)
= 5 - 12 + 7 = 0

So they are perpendicular or orthogonal.