Things at Right Angles

Introduction

Human beings since old seem to have had an affinity for things at right angles. The Pythagoras theorem is of course testament to that. And Eratosthenes’ method of measuring the radius/circumference of the earth. But why does this angle seem “right” to us? And where does it show up in our lives?

In this brief module we will examine the idea of “right-ness” in several different natural phenomena, and application areas, to develop an intuition for how this an essential property that is supremely useful.

So where do we see this idea popping up?

Physics

• In free space electromagnetic wave propagation, we have electric and magnetic fields making up the two components of the radiated wave. These are at right angles.
• An object casts a near-zero shadow when the light source illuminating it is right above it, at 90 degrees from the horizontal (i.e perpendicular).

Geometry

• Right-angled triangles and the Pythagoras theorem
• When we draw a graph in Cartesian coordinates, we wish to represent quantities \((x,y)\) on a set of axes, which are usually drawn perpendicular to each other, to ensure that the variations along each axis does not cast a shadow on the other axis, and are therefore independent.

Vectors

• When vectors are right angles, their dot-product / inner product is zero.
• When we compute the cross product / outer product of two vectors, the resultant vector is perpendicular to the plane containing the two vectors.
• If we have a family of vectors in 2 or more dimensions such that they are all mutually perpendicular to any other vector in the family, these are called an orthogonal basis set of vectors. Such vectors can be used to create a coordinate space of their own, and any other vector can be generated as a weighted sum of these basis vectors.

Waveforms

• When two (or more!) waveforms are multiplied together, and the product averaged over time, we obtain a time correlation of the two. If this happens to be zero, we classify the waveforms as being orthogonal. This is, sort of, the calculation of the “shadow” each waveform casts on the other.
• When we have a family of time functions which are each mutually orthogonal to any other waveform from the family, we have what is called an orthogonal basis set of waveforms. Such waveforms can be used to create a waveform space of their own, and any other waveform can be generated as a weighted sum of these basis waveforms.

Uses of Orthogonality

Fourier Series and Laplace Transforms

Data Visualization

Machine Learning and Deep Learning

• In a Perceptron, we have a vector dot product between the input and the weight vectors.
• In an ML-Classification problem, we consider a dataset of points containing say two classes. Each row in the data is a vector, with \(length = n(columns)\). We take a vector of weights of the same length, and take the vector dot-product of this fixed weight vector with each of the rwo-observations. The result of this operation is one dot-product number per observation in the data.
  
• Now consider the values of these dot products. Is the dot product negative, or positive in value? Can we use that polarity to decide on which observation belongs to which class? This is at the heart of an ML algorithm called Support Vector Machines.

Technology

• GPS / CDMA: See here for a quick intro to GPS. The codes used in GPS are a family of digital sequences called Gold codes. These are also an orthogonal set with near-zero cross-correlation between any pair of sequences from the set.
• Bluetooth

HR

• Brainstorming: If all members of a team sit together to brainstorm, GroupThink ensues very quickly, and people will be aligning* themselves with each other, by way of opinion or ideas.
• In a team, we need to have people who are orthogonal to each other, in the sense that they have different skill sets, and different ways of looking at the same problem. This is what makes a team effective.

References

Readings

1. Arvind V on Quora. (2016). What is orthogonality of a signal? https://qr.ae/pATe4W