Question

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Which of the following planes are parallel (or equal) to the plane x+4 y-4 z=-1 ?
(i) x-4 y-4 z=-1
(ii) -x-4 y+4 z=1
(iii) 6 x+24 y-24 z=-1
A) none of them
(B) (iii) only
(C) a

Answer 1

Two planes are parallel if their normal vectors are parallel. The normal vector of the given plane is compared to the normal vectors of the other planes to determine which ones are parallel. The answer is (iii) only.

Step-by-step explanation:

Two planes are parallel if their normal vectors are parallel. The normal vector can be found by looking at the coefficients of x, y, and z in the equation of the plane.

For the given plane x + 4y - 4z = -1, the normal vector is [1, 4, -4].

To determine which of the other planes are parallel to the given plane, we can compare their normal vectors to the normal vector of the given plane.

(i) x - 4y - 4z = -1 has a normal vector of [1, -4, -4] which is not parallel to [1, 4, -4].

(ii) -x - 4y + 4z = 1 has a normal vector of [-1, -4, 4] which is not parallel to [1, 4, -4].

(iii) 6x + 24y - 24z = -1 has a normal vector of [6, 24, -24]. Since [6, 24, -24] is a scalar multiple of [1, 4, -4], this means (iii) is parallel to the given plane.

Therefore, the answer is (C) (iii) only.