Question

What are the coordinates of the point on the plane 2latex: x+y+3z = 3 x + y + 3 z = 3 that is closest to the origin?

Answer 1

The point will be on the line through the origin that has a direction vector equal to the normal vector of the plane: (3, 1, 3).

The line's equation is ...

... (x, y, z) = t·(3, 1, 3)

The parametric equation of that line must satisfy the equation of the plane, so we can find the parameter by ...

... 3(3t) + (t) + 3(3t) = 3

... 9t +t +9t = 3

... 19t = 3

... t = 3/19

The point is (3/19)(3, 1, 3) = (9/19, 3/19, 9/19)