Final answer:
To verify if two planes are parallel, compare their normal vectors. Calculate the length of the normal vector using the coefficients of x, y, and z. Divide the absolute value of the constant term in one of the plane equations by the length of the normal vector to find the distance between the planes.
Step-by-step explanation:
To verify if two planes are parallel, we need to compare their normal vectors. The normal vector of a plane in the form ax + by + cz = d is given by the coefficients of x, y, and z. In this case, both planes have the same normal vector (3, 0, -5). Therefore, the planes are parallel.
The distance between two parallel planes can be found by dividing the absolute value of the constant term in one of the plane equations (in this case, 5) by the length of the normal vector. The length of the normal vector can be calculated using the formula sqrt(a^2 + b^2 + c^2).
Substituting the values (3, 0, -5) into the formula, we get sqrt((3)^2 + (0)^2 + (-5)^2) = sqrt(9 + 0 + 25) = sqrt(34), which is the length of the normal vector. Therefore, the distance between the planes is |5| / sqrt(34) = 5 / sqrt(34).
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