Find the exact distance  
d
  between the two parallel planes given by  
−
6
x
+
2
y
−
3
z
=
10
  and  
−
12
x
+
4
y
−
6
z
=
−
40
 .

To find the exact distance between two parallel planes, we use the formula:

\[
d = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}
\]

where the planes are given in the form:

\[
Ax + By + Cz = D_1
\]
\[
Ax + By + Cz = D_2
\]

For the given planes:

1. The first plane is: \(-6x + 2y - 3z = 10\)
2. The second plane is: \(-12x + 4y - 6z = -40\)

Let's check if the planes are parallel by comparing their normal vectors. The normal vector of the first plane is \((-6, 2, -3)\), and the normal vector of the second plane is \((-12, 4, -6)\).

Since the second normal vector is a scalar multiple of the first (specifically, \(-12, 4, -6 = 2 \times (-6, 2, -3)\)), the planes are indeed parallel.

We can rewrite the second plane in the standard form with the same normal vector as the first by dividing it by 2:

\[
-6x + 2y - 3z = -20
\]

Now, we have the planes as:

1. \(-6x + 2y - 3z = 10\) (with \(D_1 = 10\))
2. \(-6x + 2y - 3z = -20\) (with \(D_2 = -20\))

We can now apply the formula for the distance:

\[
d = \frac{|D_2 - D_1|}{\sqrt{A^2 + B^2 + C^2}}
\]

Substitute \(A = -6\), \(B = 2\), \(C = -3\), \(D_1 = 10\), and \(D_2 = -20\):

\[
d = \frac{|-20 - 10|}{\sqrt{(-6)^2 + 2^2 + (-3)^2}}
\]

Calculate step-by-step:

1. \(|D_2 - D_1| = |-20 - 10| = 30\)
2. \(\sqrt{(-6)^2 + 2^2 + (-3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7\)

Thus, the distance is:

\[
d = \frac{30}{7}
\]

So, the exact distance between the two parallel planes is:

\[
d = \frac{30}{7}
\]

Would you like more details or have any questions? 

### Relative Questions:
1. How do you determine if two planes are parallel?
2. What is the formula for the distance between two parallel planes?
3. How would the formula change if the planes were not parallel?
4. Can you find the distance between a point and a plane?
5. How do you find the angle between two planes?

### Tip:
Always check if the normal vectors of two planes are scalar multiples of each other to confirm that they are parallel before using the distance formula.

Find the exact distance d between the two parallel planes given by -6x + 2y - 3z = 10 and -12x + 4y - 6z = -40.

Learn how to find the exact distance between two parallel planes, -6x + 2y - 3z = 10 and -12x + 4y - 6z = -40. This geometry problem uses the distance formula between parallel planes, demonstrating step-by-step calculations and key vector concepts. Ideal for students in Grades 10-12.

Math Problem Statement

Find the exact distance
d between the two parallel planes given by
− 6 x + 2 y − 3 z

10 and
− 12 x + 4 y − 6 z

Solution

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