question_answer 1)

In the following exercise determine whether the given planes are parallel to perpendicular and in case they are neither, find the angle between them. (a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0 (b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0 (c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0 (d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0 (e) 4x + 8y + z – 8 = 0 and y + z – 4 = 0  

Answer:

(a) 7x + 5y + 6z + 30 = 0       And 3x ? y ? 10z 4 = 0       On comparing given planes with        A1x + b1y + c1z + d1= 0       And a2x + b2y +| c2z + d2 =0       We get, a1 = 7, b1 = 5, c1 = 6       A2 = 3, b2 = ?1, c2 = ?10       Let  be the angle between given planes.                               (b) 2x+ y + 3z ? 2 = 0 and x ? 2y + 5 = 0       On comparing given planes with a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0, we get a1 = 2, b1 = a, c1 = 3       a2 = 1, b2 = ?2, c2 = 0       As a1a2+ b1 b2 + c1c2 = 2 × 1 + 1 × (?2) + 3       × 0 = 2 ? 2 + 0 = 0        given planes are perpendicular to each other. (a)   2x ? 2y + 4z + 5 = 0 and 3x ? 3y + 6z ? 1 = 0 On comparing given planes with a1x + b1y + c1z + d1 = 0 and       a2x + b2y + c2z + d2 = 0, we get,       a1 = 2, b1 = ?2, c1 = 4       a2 = 3, b2 = ?3, c2 = 6       Here                    given planes are parallel to each other. (b)   2x ? y + 3z ? 1 = 0 and 2x ? y + 3z + 3 = 0       On comparing given ploanes with a1 + b1y + c1z d1 = 0       a2x + b2y + c2z + d2 = 0, we get, a1 = 2, b1 = ?1, c1 = 3       a2 = 2, b2 = ?1, c2 = 3       Clearly,        given planes are parallel to each other. (e)  4x + 8y + z ? 8 = 0 and y + z ? 4 = 0 On comparing given planes with a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0, we get a1 = 4, b1 = 8, c1 = 1