What is the distance between the point (2, 3, 4) and the plane x + y + z = 10? (2022)

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Q: What is the distance between the point (2, 3, 4) and the plane x + y + z = 10? (2022)

Solution: Distance = |(2 + 3 + 4 - 10) / √(1² + 1² + 1²)| = |(-1)/√3| = 1/√3.

Steps: 7

Step 1: Identify the point and the plane. The point is (2, 3, 4) and the plane equation is x + y + z = 10.
Step 2: Rewrite the plane equation in the form Ax + By + Cz + D = 0. Here, A = 1, B = 1, C = 1, and D = -10.
Step 3: Use the distance formula from a point (x0, y0, z0) to a plane Ax + By + Cz + D = 0, which is given by: Distance = |Ax0 + By0 + Cz0 + D| / √(A² + B² + C²).
Step 4: Substitute the values into the formula. For the point (2, 3, 4), we have x0 = 2, y0 = 3, z0 = 4.
Step 5: Calculate Ax0 + By0 + Cz0 + D: (1*2) + (1*3) + (1*4) - 10 = 2 + 3 + 4 - 10 = -1.
Step 6: Calculate the denominator: √(1² + 1² + 1²) = √(1 + 1 + 1) = √3.
Step 7: Plug the values into the distance formula: Distance = |-1| / √3 = 1 / √3.