The distance between the two parallel planes
`ax+by+cz+d=0`
`and ax+by+cz+d'=0`

Options

A. `|d-d^(')|/(sqrt""(a^(2)+b^(2)+c^(2)))`
B. `|d+d^(')|/(sqrt""(a^(2)+b^(2)+c^(2)))`
C. `d/(sqrt""(a^(2)+b^(2)+c^(2)))`
D. None of these

Answer

To find the distance between the two parallel planes given by the equations:

1. \( ax + by + cz + d = 0 \)
2. \( ax + by + cz + d' = 0 \)

we can follow these steps:

### Step 1: Identify the normal vector of the planes
The normal vector \( \mathbf{n} \) of the planes can be identified from the coefficients of \( x, y, z \) in the plane equations. For both planes, the normal vector is:
\[ \mathbf{n} = (a, b, c) \]

### Step 2: Use the formula for the distance between two parallel planes
The distance \( D \) between two parallel planes of the form \( ax + by + cz + d = 0 \) and \( ax + by + cz + d' = 0 \) is given by the formula:
\[ D = \frac{|d' - d|}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 3: Substitute the values into the formula
Now, we substitute \( d \) and \( d' \) into the distance formula:
\[ D = \frac{|d' - d|}{\sqrt{a^2 + b^2 + c^2}} \]

### Step 4: Simplify the expression
This expression gives us the distance between the two parallel planes in terms of the coefficients \( a, b, c \) and the constants \( d, d' \).

### Final Answer
Thus, the distance between the two parallel planes is:
\[ D = \frac{|d' - d|}{\sqrt{a^2 + b^2 + c^2}} \]

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