Show that each of the following pairs of planes are at right angles: (i) 3x + 4y – 5z = 7 and 2x + 6y + 6z + 7 = 0 (ii) x – 2y + 4z = 10 and 18x + 17y + 4z = 49
Show that each of the following pairs of planes are at right angles: (i) 3x + 4y – 5z = 7 and 2x + 6y + 6z + 7 = 0 (ii) x – 2y + 4z = 10 and 18x + 17y + 4z = 49

Answer:

(i)

if θ = 900 then cos 900 = 0

A1A2 + B1B2 + C1C2 = 0

By comparing with the standard equation of a plane,

A1 = 3, B1 = 4, C1 = -5

A2 = 2, B2 = 6, C2 = 6

LHS = A1A2 + B1B2 + C1C2

= (3 × 2) + (4 × 6) + (-5 × 6)

= 6 + 24 – 30

= 0

= RHS

The angle between the planes is 900.

Hence, proved.

(ii)

if θ = 900 then cos 900 = 0

A1A2 + B1B2 + C1C2 = 0

By comparing with the standard equation of a plane,

A1 = 1, B1 = -2, C1 = 4

A2 = 18, B2 = 17, C2 = 4

LHS = A1A2 + B1B2 + C1C2

= (1 × 18) + (-2 × 17) + (4 × 4)

= 18 + (-34) + 16

= 0

= RHS

The angle between the planes is 900.

Hence, proved.