In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (a) `7x" "+" "5y" "+" "6z" "+" "30" "=" "0` and `3x" "" "y" "" "10 z" "+" "4" "=" "0` (b) `2x" "+"

Explanation

Given plane is :
`7x+5y+6z+30 = 0` and `3z-y-10z+4=0`
Here, `a_(1)=7, b_(1) = 5, c_(1) = 6`
and `a_(2)=3,b_(2) = - 1, c_(2)= -10`
`:. a_(1)a_(2) + b_()b_(2) + c_(1)c_(2) = 7 xx 3 + 5 xx (-1) + 6 xx (-10)`
`= 21-5-6 = -44 ne 0`
Therefore, given planes are not perpendicular
Now, `(a_(1))/(a_(2)) = (7)/(3),(b_(1))/(b_(2)) = (5)/(-1) = - 5, (c_(1))/(c_(2)) = (6)/(-10) = (-3)/(5)`
Clearly, `(a_(1))/(a_(2)) ne (b_(1))/(b_(2)) ne (c_(1))/(c_(2))`.
Therefore, given planes are not parallel.
Let `theta` be the acute angle between two planes. Then
`cos theta = |(7xx3+5xx(-1)+6xx(-10))/(sqrt(7^(2)+5^(2)+6^(2))sqrt(3^(2)+(-1)^(2)+(-10)^(2)))|`
`= |(21-5-60)/(sqrt(110)sqrt(100))|`
`rArr cos theta = (44)/(110) = 2/5 rArr theta = cos^(-1)(2/5)`
Therefore, the angle between given planes is `cos^(-1)(2//5)`.
(b) Given planes are, `2x+y+3z-2 = 0`
and `x-2y+z+5=0`
Here , `a_(1)= 2, b_(1) = 1, c_(1)= 3` and `a_(2) = 1, b_(2) = -2), c_(2) = 0`
`:. a_(1)a_(2) + b_(1)b_(2)+c_(1)c_(2) = 2 xx 1 +1 xx (-2) + 3xx 0 = 0`
Therefore, given planes are perpendicular.
(c ) Given planes are
`2x-2y+4z+5 =0` and `3x-3y+6z-1=0`
Here, `a_(1) = 2, b_(1) = -2, c_(1)= 4`
and `a_(2)=3, b_(2) = - 3, c_(2)=6`
`:. a_(1)a_(2) + b_(1)b_(2) + c_(1)c_(2) = 2xx3+(-2)xx(-3)+4xx6`
`=6+6+24=36ne0`
Therefore, given planes are not perpendicular.
Now, `(a_(1))/(a_(2)) = (2)/(3), (b_(1))/(b_(2)) = (-2)/(-3) = 2/3, (c_(1))/(c_(2)) = 4/6 = 2/3`
Clearly `(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_( 2))`
Therefore, given planes are parallel.
(d). Given planes are
`2x-y+3z-1=0`
and `2x-y+3z+3 = 0`
Here, `a_(1)=2,b_(1)= -1, c_(1)= 3`
and `a_(2) = 2, b_(2)= - 1 , c_(2)= 3`
`:. a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=2xx2+(-1)xx(-1)+3xx3`
`= 4 + 1 + 9 = 14 ne 0`
Therefore, given planes are not perpendicular.
Now, `(a_(1))/(a_(2)) = 2/2 = 1, (b_(1))/(b_(2)) = (-1)/(-1) = 1, (c_(1))/(c_(2)) =3/3 = 1`
Clearly, `(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
Therefore, given planes are parallel.
(e ) Given planes are
`4x+8y+z-8=0` and `0x+1y+1z-4=0`
Here, `a_(2) = 0 , b_(2)= 1, c_(2) = 1`
`:. a_(1)a_(2)+b_(1)b_(1) +c_(1)c_(2)= 4 xx 0 + 8 xx1 + 1 xx 1`
`= 0+8+1=9 ne 0`
Therefore, given planes are not perpendicular.
Now, `(a_(1))/(a_(2)) = 4/0, (b_(1))/(b_(2)) = 8/1 , (c_1))/(c_(2)) = 1/1 = 1`
Clearly `(a_(1))/(a_(2)) ne (b_(1))/(b_(2)) ne(c_(1))/(c_(2))`
Therefore, given planes are not parallel.
Let `theta` be the acute between two planes. Then
`cos theta = |(a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)^(2))sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))|`
`= |(4xx0+8xx1+1xx1)/(sqrt(4^(2)+8^(2)+1^(2))sqrt(0^(2)+1^(2)+1^(2)))|`
`= |(9)/(9xxsqrt(2))|`
`rArr cos theta = (1)/(sqrt(2)) rArr theta = cos^(-1)(1/(sqrt(2))) = 45^(@)`