Vector, Geometric Intuition

Mathematical Vector In Geometry

Introduction

1. Contexte

What is a vector ?
From early days of school we tend to think of vector arrow on 2D grid/plane.

y
↑
│        • (3,2)
│      ↗
│    ↗
│  ↗
O────────────→ x

With notation: $\vec{u}=\left(\begin{array}{c}3\\ 2\end{array}\right)$

2. Interpretation

That is the basic overview of the subject.

But truly, a vector is less about the arrow and more about what the vector describes. If we look closely, what is actually happening in the 1st example? What does $\vec{u}$ represent?

IMPORTANT

A vector is a displacement (movement).
In our example, it represents moving 3 units along $x$ and 2 units along $y$.

• When this displacement starts from the origin $O$, we use it to define a specific position (a position vector).
• When it can start from any arbitrary point in the space¹, we call it a free vector. It represents the pure concept of movement, regardless of where you start.

Vector & Free Vector, Geometrical Intuition

1. Point vs Free Vector vs Vector in Space

A point answer the where are you are exactly in said space¹, where a Vector answer how do you move in the said space¹.

IMPORTANT

By default, the textbook/school taught to us that vector start at the origin of
space. for convenience as it’s simpler to reason about. but again a vector is movement. and any mouvement as invariable that are it’s direction and it’ magnitude.
more on that in next sections.

A free vector is a mouvement in space¹ transposed to some point in space¹.

Eg :

		   • A(2,2)
		    \ 
		     \
		      \ 
		       \
		        \
		         \> • B(4,-1)

Here we observe a free vector that we’ll name $\vec{u}$ going from the points A(2,2) to B(4,-1).
Notice that A & B are just points the actual vector vector is how do we get from a A to B ?

Simple : A + (2, -3)
So our vector’s $\vec{u}=\left(\begin{array}{c}2\\ -3\end{array}\right)$

IMPORTANT

I insist $\vec{u}$ is pure movement, not a value, it’s transformation that can be realized at any point in space¹.

As well it’s symmetrical meaning we can B - $\vec{u}$ = A

2. Characteristic of a Vector

2.1 Geometric Vector

A vector or more precisely in this case a geometric vector that characterized by a :

• a sens
• an angle
• a direction
• and a magnitude

it’s important to differentiate a geometric vector, from non geometric one. reason being that not every space require all characteristic given to vector.

Footnotes

1. Intuition (State/Parameter Space): Much like a sample space $\Omega$ in probability theory represents the universe of all possible outcomes, a vector space here acts as the universe of all possible parameter configurations. Each vector is a specific “point” or snapshot in this universe, uniquely defined by a tuple of parameters (coordinates) spanning across the available dimensions. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶