verify that the two planes are parallel, and find the distance between the planes. (round your answer to three decimal places)

Question

2x-5z=5
2x-5z=6

Answer 1

To verify that the two planes are parallel, we need to check if their normal vectors are parallel. The normal vector of a plane with the equation of the form ax + by + cz = d is (a, b, c).

Both planes have the same normal vector (2, 0, -5), so they are parallel.

To find the distance between the planes, we can choose a point on one of the planes and find the distance between this point and the other plane.

Let's choose the point (0, 0, 1) on the plane 2x - 5z = 5. The distance between this point and the plane 2x - 5z = 6 is:

|2(0) - 5(1) - 6| / sqrt(2^2 + 0^2 + (-5)^2) = |-11| / sqrt(29) = 11/sqrt(29)

So, the distance between the two planes is approximately 1.999 units when rounded to three decimal places.