Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z

Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)

Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)

Step 1: Identify the equations of the two parallel planes. They are 2x + 3y + 4z = 5 and 2x + 3y + 4z = 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, and D is a constant.
Step 3: From the equations, we see that A = 2, B = 3, C = 4 for both planes.
Step 4: Identify the constants D1 and D2 from the plane equations. Here, D1 = 5 and D2 = 10.
Step 5: Use the formula for the distance between two parallel planes: Distance = |D1 - D2| / √(A² + B² + C²).
Step 6: Calculate the absolute difference between D1 and D2: |5 - 10| = 5.
Step 7: Calculate the denominator: √(A² + B² + C²) = √(2² + 3² + 4²) = √(4 + 9 + 16) = √29.
Step 8: Substitute the values into the distance formula: Distance = 5 / √29.
Step 9: The final answer is the distance between the two parallel planes.

Distance Between Parallel Planes – The formula for the distance between two parallel planes of the form Ax + By + Cz = d is given by |d1 - d2| / √(A² + B² + C²).
Understanding Plane Equations – Recognizing that the coefficients of x, y, and z in the plane equations are the same indicates that the planes are parallel.