What is the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10? (2023)

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Q: What is the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10? (2023)

Solution: Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.

Steps: 7

Step 1: Identify the equations of the two parallel planes. They are x + 2y + 3z = 4 and x + 2y + 3z = 10.
Step 2: Recognize that the general form of a plane is Ax + By + Cz = D, where A, B, and C are the coefficients of x, y, and z respectively.
Step 3: From the equations, identify A = 1, B = 2, C = 3 for both planes.
Step 4: Identify the values of D for each plane. For the first plane, D1 = 4, and for the second plane, D2 = 10.
Step 5: Calculate the absolute difference between D1 and D2. This is |D1 - D2| = |4 - 10| = 6.
Step 6: Calculate the denominator, which is the square root of the sum of the squares of A, B, and C. This is √(A² + B² + C²) = √(1² + 2² + 3²) = √(1 + 4 + 9) = √14.
Step 7: Finally, calculate the distance between the two planes using the formula: Distance = |D1 - D2| / √(A² + B² + C²). Substitute the values: Distance = 6 / √14.