Calculating the Perpendicular Vector to a Plane: Step-by-Step Guide

Finding a Vector Perpendicular to a Plane

A vector perpendicular to a plane, also known as the normal vector, is crucial in many fields, including geometry, physics, and computer graphics. Here’s how you can find such a vector using different methods.

1. Using Three Points

Given three points, A(x₁, y₁, z₁), B(x₂, y₂, z₂), C(x₃, y₃, z₃) on the plane, follow these steps:

Step 1: Find Vectors AB and AC

AB = (x2 - x1, y2 - y1, z2 - z1)

AC = (x3 - x1, y3 - y1, z3 - z1)

Step 2: Cross Product

Find the cross product AB × AC to get the perpendicular vector:

u × v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)

Example:

Given points P(2, 0, -1), Q(1, 1, 3), R(0, -1, 2):

• Find vectors PQ = (-1, 1, 4) and PR = (-2, -1, 3).
• Cross product: PQ × PR = (7, -5, 3)

So, the vector perpendicular to the plane is (7, -5, 3).

2. From Plane Equation

For a plane equation ax + by + cz = d, the coefficients a, b, c form the normal vector.

Example:

For the plane 3x + y - 2z = 12, the normal vector is (3, 1, -2).

3. Finding the Unit Vector

To get a unit vector perpendicular to the plane, normalize the normal vector by dividing it by its magnitude:

n̂ = (a, b, c) / √(a² + b² + c²)

Example:

For the normal vector (5, 1, -3):

n̂ = (5, 1, -3) / √(5² + 1² + (-3)²) ≈ (5/√35, 1/√35, -3/√35)

Practical Applications

• Engineering: To determine forces perpendicular to a surface.
• Physics: In calculating normal vectors for fields like electromagnetism.
• Computer Graphics: To compute surface normals, essential for lighting and rendering.

Conclusion

Finding a vector perpendicular to a plane is essential across various domains. Whether you’re using points on the plane or a plane equation, the steps above help calculate the normal vector efficiently.