Two planes are perpendicular if and only if their normal vectors are orthogonal. Remember that if a plane is in the form of ax+by+cz+d=0, then its normal vector is ⟨a,b,c⟩. Also remember that two vectors are orthogonal if and only if their dot product is equal to 0.
Problem: Determine if the planes 5x+4y−2z+13=0 and 4x−2y+6z−1=0 are perpendicular.
Solution: The normal vector of the first plane is ⟨5,4,−2⟩, and the normal vector of the second plane is ⟨4,−2,6⟩. Their dot product is
⟨5,4,−2⟩⋅⟨4,−2,6⟩=5⋅4+4⋅(−2)+(−2)⋅6=20−8−12=0
Since their dot product is 0, the two planes are perpendicular.
Problem: Determine if the planes x+2y−3z−4=0 and −4x−3y+2z+1=0 are perpendicular.
Solution: The normal vectors are ⟨1,2,−3⟩ and ⟨−4,−3,2⟩. Their dot product is
⟨1,2,−3⟩⋅⟨−4,−3,2⟩=1⋅(−4)+2⋅(−3)+(−3)⋅2=−4−6−6=−16
Since −16=0, the planes are not perpendicular.
