The planes 2x - y + 4z = 3 and 5x - 2.5y + 10z = 6 are :

Options

A. perpendicular
B. parallel
C. intersect along y - axis
D. passes through `(0,0,(5)/(4))`.

Answer

To determine the relationship between the two planes given by the equations \(2x - y + 4z = 3\) and \(5x - 2.5y + 10z = 6\), we can follow these steps:

### Step 1: Identify the normal vectors of the planes
The general equation of a plane can be expressed as \(Ax + By + Cz = D\), where the vector \((A, B, C)\) is the normal vector of the plane.

For the first plane \(P_1: 2x - y + 4z = 3\):
- The coefficients give us the normal vector \(n_1 = (2, -1, 4)\).

For the second plane \(P_2: 5x - 2.5y + 10z = 6\):
- The coefficients give us the normal vector \(n_2 = (5, -2.5, 10)\).

### Step 2: Check if the planes are parallel
Two planes are parallel if their normal vectors are scalar multiples of each other. We can check this by comparing the ratios of the components of the normal vectors.

Calculating the ratios:
\[ \frac{n_1}{n_2} = \left(\frac{2}{5}, \frac{-1}{-2.5}, \frac{4}{10}\right) = \left(\frac{2}{5}, \frac{2}{5}, \frac{2}{5}\right) \]

Since all components yield the same ratio, \(n_1\) and \(n_2\) are indeed scalar multiples of each other. Therefore, the planes are parallel.

### Step 3: Conclusion
Since the normal vectors are scalar multiples, we conclude that the planes \(P_1\) and \(P_2\) are parallel.

### Final Answer
The planes \(2x - y + 4z = 3\) and \(5x - 2.5y + 10z = 6\) are parallel.

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