3.4.4 Normal Vector

A vector that expresses the vertical direction of a surface is called a normal vector:

How to Calculate a Normal Vector

To calculate the normal vector of a surface, you need to find what is called the outer product of the two vectors that specify that surface. You can use the following step-by-step technique to calculate the outer product of two vectors
and where:

1. Arrange and write down the coordinates of each vector and as two lines like this:
   
   
   
   
   
2. Repeat each X coordinate of and on the far right end of each line like this:
   
   
   
   
   
3. Draw connecting lines like this:
   
   
   
   
4. Calculate , , and using these formulas:
   
   
   
   
   
   is the X coordinate of the result, is the Y coordinate, and is the Z coordinate.
   
   

In summary, to obtain the outer product of vectors and , use this formula:

Example Calculation

Here is a second example that shows how to find the normal vector N of the plane defined by the three points A, B and P shown in this illustration:

Note: When working with the two vectors and , you must be very careful to factor in each vector's direction.

Calculate the normal vector N by using this formula:

To get the outer product of the vectors, apply these formulas:

Because and are vectors, specify each coordinate component like this:

Therefore, the final detailed formula for the normal vector N (Nx,Ny,Nz) is:

You need the outer product for many geometric calculations, not just to calculate the normal vector.

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Last Updated March, 1999