Question

Which of the following planes are parallel (or equal) to the plane x-4 y+4 z=3 ?

(i) x+4 y+4 z=3
(ii) -x+4 y-4 z=-3
(iii) 5 x-20 y+20 z=3
(A) (i) and (iii) only (B) (i) only (C) (i) and (ii) only (D) all of them (E) (iii) only (F) none of them (G) (ii) only (H) (ii) and (iii) only

Answer 1

Two planes are parallel or equal if their normal vectors are scalar multiples of each other. Only plane (iii) 5x - 20y + 20z = 3 has a normal vector that is a scalar multiple of the normal vector of the given plane x - 4y + 4z = 3. Thus, the correct answer is (E) (iii) only.

Step-by-step explanation:

To determine which planes are parallel or equal to the given plane x - 4y + 4z = 3, we need to look at the normal vectors of these planes. Two planes are parallel if their normal vectors are scalar multiples of each other.

The normal vector for the given plane is (1, -4, 4).

For plane (i) x + 4y + 4z = 3, the normal vector is (1, 4, 4), which is not a scalar multiple of the given plane's normal vector.

For plane (ii) -x + 4y - 4z = -3, the normal vector is (-1, 4, -4), which is also not a scalar multiple of the given plane's normal vector.

For plane (iii) 5x - 20y + 20z = 3, the normal vector is (5, -20, 20), which is a scalar multiple of the given plane's normal vector. Specifically, it's 5 times the given plane's normal vector.

Therefore, the planes that are parallel or equal to the given plane x - 4y + 4z = 3 are plane (iii) only. The correct answer to the question is (E) (iii) only.