Prove that the two planes, 3x - 4y + 5z = 0 and 2x - y - 2z = 5 are mu

1.	Prove that the two planes, 3x - 4y + 5z = 0 and 2x - y - 2z = 5 are mutually perpendicular.
Answer» Given equations, 3x - 4y + 5z = 0 ...(i) 2x - y - 2z = 0 ...(ii) According to formula, a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is perpendicular if, a₁a₂ + b₁b₂ + c₁c₂= 0 Here, a₁= 3, b₁ = -4, c₁ = 5 a₂ = 2, b₂ = -1, c₂ = -2 then a₁a₂ + b₁b₂ + c₁c₂ = 3 x 2 + (-4) x (-1) + 5 x (-2) = 6 + 4 - 10 = 10 - 10 = 0 Hence, eq. (i) and (ii) is perpendicular.

Prove that the two planes, 3x - 4y + 5z = 0 and 2x - y - 2z = 5 are mutually perpendicular.

Answer»

Given equations,

3x - 4y + 5z = 0 ...(i)

2x - y - 2z = 0 ...(ii)

According to formula, a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is perpendicular if,

a₁a₂ + b₁b₂ + c₁c₂= 0

Here, a₁= 3, b₁ = -4, c₁ = 5

a₂ = 2, b₂ = -1, c₂ = -2

then a₁a₂ + b₁b₂ + c₁c₂

= 3 x 2 + (-4) x (-1) + 5 x (-2)

= 6 + 4 - 10 = 10 - 10 = 0

Hence, eq. (i) and (ii) is perpendicular.