Question

Use Lagrange multipliers to find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 3). (x, y, z) =

Answer 1

To find the closest point on the plane x - 2y + 3z = 6 to the point (0, 1, 3) using Lagrange multipliers, set up and solve a system of equations derived from the gradient of the distance function and the gradient of the constraint equation.

Step-by-step explanation:

To use Lagrange multipliers to find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 3), we must minimize the distance function D(x, y, z) = (x-0)² + (y-1)² + (z-3)², subject to the constraint x - 2y + 3z = 6. The method of Lagrange multipliers involves finding the scalar λ (the Lagrange multiplier) such that the gradient of D equals λ times the gradient of the constraint.

Steps:

1. Compute the gradient of the distance function: ∇D = (2(x-0), 2(y-1), 2(z-3)).
2. Compute the gradient of the constraint: ∇g = (1, -2, 3).
3. Set the gradient of D equal to λ times the gradient of g, resulting in the system of equations:

• 2(x-0) = λ(1)
• 2(y-1) = -λ(2)
• 2(z-3) = λ(3)
• x - 2y + 3z = 6 (the constraint)

Solve this system of equations for x, y, z and λ.

Upon solving the equations, we find the closest point on the plane to (0, 1, 3), represented by the coordinates (x, y, z).