Analytic Geometry Part 3 Reducing Dimensionality

For this chapter, we will reduce three-dimensional data to one-dimensional data using the techniques of projection and rotation. The (x, y, z.) data will be projected onto the (x, z) plane and then rotated onto the x axis. This chapter is purely pedagogical and is intended only to demonstrate the use of projection and rotation as geometric terms. 

The exercise for this column is to reduce a point on a vector in 3-D space to a point on a vector in 2-D space, then to further reduce the point on a vector in 2-D space to a point on a vector in 1-D space - all the while maintaining as much information as possible. So (x, y, z) is reduced to (x, z), which is further reduced to (x). This process can be represented in symbolic language as (x, y, z) (x, z.) - (x). 

Because the third dimension is represented by the z axis, we calculate the z-direction angle on the (x, z) plane as y  

To calculate the length of the horizontal vector for the projection of vector AB onto the (x, z) plane, we can use 

In our next chapter, we will be applying the lessons reviewed over these past three chapters toward a better understanding of the geometric concepts relative to multivariate regression. 

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