Vectors

Introduction

What might these figures betoken??

(a) A

(b) B

Figure 1: Flow Field: Vectors at each point in a Plane

Discussion #1

Each figure¹ shows a bunch of arrows, centered at different locations on a regular grid. What sort of quantities could these arrows describe?

Clearly the arrows’ size and their direction has something to do with how they represent some underlying phenomena.

Could Figure 1 (a) represent say wind flow over a region? Yes the air is quite still around (0, 0), but picks up in force and also angle as we move rightwards. If you were going to fly a kite with this wind pattern, where would you stand, and where would you point your kite at?

And could Figure 1 (b) represent say the flow of water in a whirlpool? Can you swim?

Discussion #2

• Each of these arrows is a vector: the arrows show both the force magnitude and its direction ## Wait, But Why?

• We need Quantities that have both magnitude and direction, together at the same time. Think force, velocity, acceleration, momentum, torque, magnetic field, electric field, etc.

• A vector is such a quantity.

• With two components ( at least ) a vector is able to achieve this dual purpose at the same time! It is a terrific Mathematical Tool!

References

1. https://www.math.hkust.edu.hk/~machas/vector-calculus-for-engineers.pdf

Footnotes

1. Data Visualization geeks will call this a scatter plot with arrows used instead of points.