Multivariable Calculus: Find the point on the sphere closest to a plane

We look for closest point on the unit sphere to the plane 3x + y + 4z = 10 using a geometric approach. We start by sketching the sphere and the plane. Our analysis reveals that the shortest line segment from the sphere to the plane is perpendicular to the plane and parallel to the plane's orthogonal vector, (3, 1, 4). We find that the closest point on the sphere is the one where its tangent plane aligns with this vector. After normalizing the vector (3, 1, 4) to unit length, we determine that the closest point is in the first octant and confirm this with visualization.

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