Question

Use Lagrange multipliers to find the point on the plane

x−2y+3z=6 that is closest to the point (0, 3, 2).
a) (0, 2, 0)
b) (1, 1, 1)
c) (2, 0, 2)
d) (3, 1, 1)

Answer 1

To find the closest point on the plane x - 2y + 3z = 6 to the point (0, 3, 2), we can use Lagrange multipliers. The closest point is (0, 2, 0).

Step-by-step explanation:

To find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 3, 2), we can use Lagrange multipliers. Let's define the distance between the two points as D(x, y, z) = (x - 0)^2 + (y - 3)^2 + (z - 2)^2. Our objective is to minimize D(x, y, z) subject to the constraint x - 2y + 3z = 6.

We can set up the Lagrange function: L(x, y, z, λ) = D(x, y, z) - λ(x - 2y + 3z - 6). To find the closest point, we need to find the values of x, y, z, and λ that minimize L(x, y, z, λ).

Taking partial derivatives of L with respect to x, y, z, and λ, and setting them equal to zero, we can solve for x, y, z, and λ. After solving the system of equations, we find that the closest point on the plane is (0, 2, 0).