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

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

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

Step 1: Identify the equations of the two parallel planes. The first plane is x + 2y + 3z = 4 and the second plane is x + 2y + 3z = 10.
Step 2: Recognize that the distance between two parallel planes can be calculated using the formula: Distance = |d1 - d2| / √(a² + b² + c²), where d1 and d2 are the constant terms from the plane equations.
Step 3: Identify d1 and d2 from the plane equations. Here, d1 = 4 and d2 = 10.
Step 4: Calculate the absolute difference between d1 and d2: |4 - 10| = | -6 | = 6.
Step 5: Identify the coefficients a, b, and c from the plane equations. In this case, a = 1, b = 2, and c = 3.
Step 6: Calculate the value of √(a² + b² + c²): √(1² + 2² + 3²) = √(1 + 4 + 9) = √14.
Step 7: Substitute the values into the distance formula: Distance = 6 / √14.
Step 8: The final answer for the distance between the two parallel planes is 6 / √14.

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.