Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)

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Q: Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)

Solution: Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.

Steps: 8

Step 1: Identify the equations of the two parallel planes. The first plane is 2x + 3y + z = 5 and the second plane is 2x + 3y + z = 10.
Step 2: Recognize that the general form of a plane is Ax + By + Cz = d, where A, B, C are the coefficients of x, y, z respectively, and d is a constant.
Step 3: From the equations, identify A, B, C, and the constants d1 and d2. Here, A = 2, B = 3, C = 1, d1 = 5, and d2 = 10.
Step 4: Use the formula for the distance between two parallel planes: Distance = |d1 - d2| / √(A² + B² + C²).
Step 5: Calculate the absolute difference between d1 and d2: |5 - 10| = 5.
Step 6: Calculate the denominator: √(A² + B² + C²) = √(2² + 3² + 1²) = √(4 + 9 + 1) = √14.
Step 7: Substitute the values into the distance formula: Distance = 5 / √14.
Step 8: The final answer is the distance between the two parallel planes.