Determine if the normal vector to a plane can be derived from the closest point to the origin by normalizing its coordinates | Step-by-Step Solution

1. What We're Solving:

We need to determine whether the normal vector to a plane can be found by taking the coordinates of the closest point on the plane to the origin and normalizing them (dividing by the point's magnitude). This is a beautiful question that connects several geometric concepts!

2. The Approach:

We'll use the fundamental relationship between a plane, its normal vector, and the closest point to the origin. The key insight is understanding what "closest point" means geometrically and how it relates to perpendicularity.

3. Step-by-Step Solution:

Step 1: Visualize the setup Imagine a plane in 3D space and the origin (0,0,0). The closest point on the plane to the origin is special - it's where the shortest distance from origin to plane occurs.

Step 2: Key geometric insight The shortest path from the origin to any plane is always perpendicular to that plane! This means the line segment from the origin to the closest point on the plane is perpendicular to the plane.

Step 3: What does "perpendicular to the plane" mean? If a line is perpendicular to a plane, then that line has the same direction as the plane's normal vector. This is crucial!

Step 4: Connect the closest point to the normal vector Let's call the closest point P = (x₀, y₀, z₀). Since the line from origin O = (0,0,0) to P is perpendicular to the plane, the vector OP⃗ = (x₀, y₀, z₀) points in the same direction as the normal vector.

Step 5: From direction to unit normal The vector (x₀, y₀, z₀) points in the right direction, but it might not have length 1. To get a unit normal vector (magnitude = 1), we normalize it:

Normal vector = (x₀, y₀, z₀) / √(x₀² + y₀² + z₀²)

This is exactly what the question asks about!

Step 6: Why this works mathematically If we write the plane equation as ax + by + cz = d (where (a,b,c) is the normal vector), the closest point to origin has coordinates that are proportional to (a,b,c). The normalization gives us the unit normal.

4. The Answer:

Yes! The normal vector to a plane equals the coordinates of the closest point to the origin, normalized by its magnitude. This works because the line from the origin to the closest point on the plane is always perpendicular to the plane, which means it points in the direction of the normal vector.

5. Memory Tip:

Remember: "The shortest path teaches the normal direction!" The closest point creates the shortest path from origin to plane, and this shortest path is always perpendicular to the plane, giving you the normal direction. Just normalize it to get a unit normal vector!

This is such an elegant result - it shows how geometric intuition (shortest distance) directly gives us algebraic tools (normal vectors)! 🎯