Q.

1.(i) Find the angle between the planes x+2y+2z5=0 and 3x+3y+2z8=0.
(ii) Find the angle between the planes 2xy+z6=0 and x+y+2z7=0

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Detailed Solution

(i) Given planes are x + 2y + 2z – 5 = 0 ….(1)
a1=1,b1=2,c1=2and 3x+3y+2z8=0 ...(2)a2=3,b2=3,c2=2
Let θ is the angle between  the planes (1) & (2) then
cosθ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22cosθ=|1(3)+2(3)+2(2)|12+22+2232+32+22              =|3+6+4|922              =13322θ=cos113322
(ii) Given planes are
2xy+z6=0(1)a1=2,b1=1,c1=1x+y+2z7=0(2)a2=1,b2=1,c2=2
Let θ is the angle between  the planes (1) & (2) then
cosθ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22            =|21+2|22+(1)2+1212+12+22           =366            =36=12cosθ=12θ=π3

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