Vectors: ℝⁿ Dimensions

Vectors and Motion

Position Point

A point exists even if nothing stands on it. It does not move, as it is space itself. It is a type of abstraction that allows us to separate and differentiate spaces. We can do that by structuring its Dimensions. These are components that, ordered together, establish properties of these Points. It can be real Spaces with real Dimensions, but it may also be Data Points of abstract objects (such as '❨height,weight❩' represents a person's length-weight space).

Point: A location in a space.

A point $𝙿$ may be described by its basic components, its specific datum of the space. These components exist in spaces that do not mix together (they are perpendicular to each other), but exist in an ordered Cartesian Product macro-space that they build by being together.

\color{#00b4d8}
\text{𝙿osition} = 𝚙_x ∊ 𝕏 ⟂ 𝚙_y ∊ 𝕐 ⟂ 𝚙_z ∊ ℤ

\color{#00b4d8}
 = （𝚙_x，𝚙_y，𝚙_z）∈ (𝕏 ⨯ 𝕐 ⨯ ℤ)

\begin{matrix}
• & ● & •  & • & • & • & •\\
• & ● & •  & • & • & • & •\\
𝚙_𝑦 & {\color{#00b4d8}⦿} & •  & •  & {\color{#00b4d8}𝚙◉} & • & •\\
• & ● & •  & • & •  & • & •\\
• & ● & •  & • & • & • & •\\
• & ● & •  & • & • & •  & •\\
● & ⌱ & ●  & ● & {\color{#00b4d8}⦿} & ●  & ● \\
• & ● & •  & • & 𝚙_𝑥 & • & •\\


\end{matrix}

Points do not mix together, they just exist there. But there are also other types of entities that belong to those spaces. When we transport something from point to point, compare them, or interact within the framework of these points we are using Vectors.

Vector: defined by a magnitude and a direction.

For example, when we "locate" a point first we establish a "null state", an "origin" from where all other points are synthesized against. We can locate points by the use of the Position Vector of that data point by adding the component's values from the "null".

\color{#00b4d8}
\vec{\text{position}} = 𝚙_x · 𝑥̂ + 𝚙_y · 𝑦̂ + 𝚙_z · 𝑧̂

\color{#00b4d8}
 = ⟨𝚙_x，𝚙_y，𝚙_z⟩

This represents the position vector in three-dimensional space, expressed in terms of its components along the $x$, $y$, and $z$ axes.

\begin{matrix}
• & ● & •  & • & • & • & •\\
• & ● & •  & • & • & • & •\\
• & {\color{#00b4d8}▲𝚙_𝑦} & •  & •  & {\color{#00b4d8}\vec{𝚙}◥} & • & •\\
• & ● & •  & • & •  & • & •\\
• & ● & •  & • & • & • & •\\
• & ● & •  & • & • & •  & •\\
● & ⌱ & ●  & ● & {\color{#00b4d8}𝚙_𝑥►} & ●  & ● \\
• & ● & •  & • & • & • & •\\
\end{matrix}

If we apply a vector to a point, by adding, we obtain a different point in the data space.

\color{Cerulean} 
 𝚙_i +\vec{u} = 𝚙_f 

\color{Cerulean} 
[ℙ] + [\vec{𝕍}] = [ℙ]

Therefore we can establish the transformation of the vector as the difference between these point elements.

\color{Cerulean} 
\vec{u} = 𝚙_f -  𝚙_i 

\color{Cerulean} 
[\vec{𝕍}] = [ℙ] - [ℙ] 

Notice that the "position" of origin of the vector doesn't define it. The same vector may exist in any and all positions of the data space. A vector is, therefor, an application on the data, not a data point. Actions, not objects. They are verbs in the language of math.

Vectors are morphisms.

Magnitude of a Vector $\vec{u}$

A vector represents a change. Its magnitude is the energy of that change: how far, how much, how strongly it acts. Its magnitude, or length, is the measure of movement it produces and the extent of transformation it represents, without a specific direction.

With this formula we calculate the length of a vector in three-dimensional space, using the old Pythagorean theorem.

\color{Cerulean} 
|𝐮⃗| = \sqrt{u_x^2 + u_y^2 + u_z^2} = \sqrt{𝐮⃗·𝐮⃗} = 𝑈

A point is a place.
A vector is an arrow.
But the magnitude? That is the tension in the bow.

Unit Vector in the Direction of $\vec{u}$

A unit vector is a minimal vector distilled to its pure direction. It moves in a certain direction, but only as far as one step.

To create it, we reduce to a unit its magnitude:

\color{BlueGreen} 
\hat{u} = \frac{𝐮⃗}{|𝐮⃗|}

A unit vector has a magnitude of 1 and indicates the direction of $𝐮⃗$. It is obtained by dividing the vector by its magnitude.

Therefore we can also express a vector as:

\color{Cerulean} 
𝐮⃗ ＝ 𝑈û = û𝑈

That is, a magnitude $U$ in the direction $û$.

Vector Algebra

These are the fundamental rules of vector addition, subtraction, and scalar multiplication, highlighting properties like commutativity and distributivity.

\color{Emerald}
\begin{matrix}
𝐮⃗ + 𝐯⃗ = 𝐯⃗ + 𝐮⃗              && (commutative) \\
𝐮⃗ + (𝐯⃗ + 𝐰⃗) = (𝐮⃗ + 𝐯⃗) + 𝐰⃗ && (associative)\\
𝐮⃗ + 0⃗ = 𝐮⃗                 && (identity)\\
𝐮⃗ - 𝐮⃗ = 0⃗                 && (inverse)\\
c(𝐮⃗ + 𝐯⃗) = c𝐮⃗ + c𝐯⃗         && (scalar distributive over +)\\
(c + d)𝐮⃗ = c𝐮⃗ + d𝐮⃗         && (distributive over scalars)\\
(cd)𝐮⃗ = c(d𝐮⃗) = d(c𝐮⃗)      && (scalar associativity)\\
1𝐮⃗ = 𝐮⃗                     && (unit scalar)\\
0𝐮⃗ = 0⃗                    && (zero scalar)\\
c0⃗ = 0⃗                   && (zero vector)
\end{matrix}

Commutative:

graph LR
  Start@{ shape: start, stroke: black } 
  PV@{ shape: start, stroke: blue } 
  PU@{ shape: start, stroke: cyan } 
  PUV@{ shape: start, stroke: purple } 

  Start VectorU1@--->|𝑢⃗| PU
  PU VectorV1@-->|𝑣⃗| PUV

  Start VectorUV@--->|"𝑢⃗+𝑣⃗"| PUV

  Start VectorV2@-->|𝑣⃗| PV
  PV VectorU2@--->|𝑢⃗| PUV

  classDef U stroke: Lime, stroke-width: 3;
  classDef V stroke: Red, stroke-width: 3;
  classDef UV stroke: Gold, stroke-width: 3;
  class VectorU1 U;
  class VectorU2 U;
  class VectorV1 V;
  class VectorV2 V;
  class VectorUV UV;

Associative

graph LR
  Start@{ shape: start, stroke: black } 
  PU@{ shape: start, stroke: cyan } 
  PUV@{ shape: start, stroke: blue } 
  PUVW@{ shape: start, stroke: cyan } 
  PUV@{ shape: start, stroke: purple } 

  Start VectorU@--->|𝑢⃗| PU


  PU VectorV@-->|𝑣⃗| PUV
  PU VectorVW@-->|"𝑣⃗+𝑤⃗"| PUVW
  PUV VectorW@-->|𝑤⃗| PUVW

  Start VectorUV@--->|"𝑢⃗+𝑣⃗"| PUV

  Start VectorUVW@--->|"𝑢⃗+𝑣⃗+𝑤⃗"| PUVW

  classDef V stroke: Red, stroke-width: 3;
  classDef C stroke: Tomato, stroke-width: 3 , stroke-dasharray: 4 8;
  classDef D stroke: Crimson, stroke-width: 3 , stroke-dasharray: 8 10;
  class VectorU V;
  class VectorUV C;
  class VectorUVW D;
  class VectorV V;
  class VectorW V;
  class VectorVW C;

Vector Difference

\color{Lime}
\vec{P_1P_2} = ⟨x_2 - x_1，y_2 - y_1，z_2 - z_1⟩

This formula defines a vector as the difference between two points in space.

Distance between $𝖯$ and $𝖰$

\color{Lime}
\text{distance}(𝖯_1, P_2) = |\vec{P_1P_2}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

This gives the Euclidean distance between two points $𝖯_1$ and $P_2$ in three-dimensional space.

In ℝ²

If a vector $𝐮⃗$ is represented in the plane by an Origin Point $\color{Red}𝖯(x_1, y_1)$ and an endpoint $\color{Red}𝖰(x_2，y_2)$, then:

\color{Gold}
𝐮⃗ = ⟨x_2-x_1，y_2-y_1⟩

The components of $𝐮⃗$ are found by subtracting the coordinates of the origin from those of the endpoint.

Magnitude in ℝ²

\color{DarkOrange}
|𝐮⃗| = \sqrt{a^2 + b^2}

The Magnitude or length of a vector $\color{DarkOrange} 𝐮⃗ = ⟨a，b⟩$.

Vector Algebra

\color{Gold}
\begin{matrix}
𝐮⃗ + 𝐯⃗ = ⟨a_u + a_v, b_u + b_v⟩ \\
￢𝐮⃗ = ⟨ ￢a_u, ￢b_u⟩ \\
𝐮⃗ - 𝐯⃗ = ⟨a_u - a_v, b_u - b_v⟩ \\
c𝐮⃗ = ⟨c·a_u, c·b_u⟩
\end{matrix}

These formulas describe addition, negation, subtraction, and scalar multiplication of vectors in two dimensions.

Unitary Vectors in ℝ²

\color{Gold}
\begin{matrix}
𝐢̂ = ⟨1，0⟩ \quad 𝐣̂ = ⟨0，1⟩ \\
𝐮⃗ = a_u𝐢̂ + b_u𝐣̂ \\
𝐮⃗ = |𝐮⃗| \cos𝜃 ·𝐢̂ + |𝐮⃗| \sin𝜃 ·𝐣̂
\end{matrix}

The unit vectors $𝐢̂$ and $𝐣̂$ define the standard basis for ℝ², and any vector can be expressed in terms of these basis vectors. They form the 'alphabet' to construct vector 'words'.

Dot Product in ℝ²

\color{Gold}
\begin{matrix}
𝐮⃗ · 𝐯⃗ = a_u · a_v + b_u · b_v \\
𝐮⃗ · 𝐯⃗ = |𝐮⃗||𝐯⃗|\cos𝜃 \\
\end{matrix}

This gives:

\color{Gold}
\begin{matrix}
\cos𝜃 = \frac{𝐮⃗ · 𝐯⃗}{|𝐮⃗||𝐯⃗|} \\
𝐮⃗ · 𝐮⃗ = |𝐮⃗|^2
\end{matrix}

The dot product measures how much one vector extends in the direction of another and is used to find angles between vectors. It gives a measure of similarity or opposition.

Two vectors are perpendicular (they do not mix or oppose at all) if:

\color{Gold}
𝐮⃗ \perp 𝐯⃗ \iff 𝐮⃗ · 𝐯⃗ = 0

Two vectors are perpendicular if their dot product is zero.

Component of $𝐮⃗$ along $𝐯̂$

\color{Gold}
𝐮⃗_{𝐯⃗} = \frac{𝐮⃗ · 𝐯̂}{|𝐯⃗|} \cdot 𝐯̂

This gives the projection (a shadow casted perpendicular) of one vector onto the direction of another.

Pro

\color{gold}
\begin{matrix}
𝐴_ʙ = 𝐴 cos𝜗 = 𝐴⃗·B̂ \\ &&
   𝐴⃗_ʙ=𝐴_ʙ B̂ = (𝐴⃗·B̂)B̂ \\
𝜗 = 𝐴∡𝐵
\end{matrix}

Cross Product in ℝ³

\color{Yellow}
\begin{matrix}
𝐮⃗ × 𝐯⃗ = \text{det}
\begin{bmatrix}
\hat{x} & \hat{y} & \hat{z} \\
u_x & u_y & u_z \\
v_x & v_y & v_z
\end{bmatrix} \\
= ⟨u_y v_z - u_z v_y，u_z v_x - u_x v_z，u_x v_y - u_y v_x⟩
\end{matrix}

The cross product produces a vector orthogonal to both input vectors and is proportional to the area of the parallelogram spanned by them.

Triple Product

This scalar is the volume of a parallelepiped with the input Position Vectors as edges.

\color{Ruby}
𝐮⃗ · (𝐯⃗ × 𝐰⃗) = \text{Volume of parallelepiped formed by } 𝐮⃗, 𝐯⃗, 𝐰⃗

\color{Ruby}
𝐮⃗ · (𝐯⃗ × 𝐰⃗) = \text{det}
\begin{bmatrix}
u_x & u_y & u_z \\
v_x & v_y & v_z \\
w_x & w_y & w_z
\end{bmatrix}

This expresses the component of $𝐮⃗$ along $𝐯⃗$, giving its projection as a vector.

Triangle Inequality

For any two vectors $𝐮⃗$ and $𝐯⃗$,

⎮𝐮⃗ + 𝐯⃗⎮ ≤ ⎮𝐮⃗⎮ + ⎮𝐯⃗⎮

Equality holds if and only if either $(𝐯⃗)$ is a scalar multiple of $(𝐮⃗)$, or one of them is 0.

The shortest path is never longer than the joined paths.

Cauchy-Schwarz Inequality

For any two vectors $𝐮⃗$ and $𝐯⃗$,

⎮𝐮⃗ · 𝐯⃗⎮ ≤ ⎮𝐮⃗⎮ · ⎮𝐯⃗⎮

This can be seen in 2D simply as:

⎮⟨𝑎,𝑏⟩ · ⟨𝚙,𝚚⟩⎮ ≤ ⎮⟨𝑎,𝑏⟩⎮ · ⎮⟨𝚙,𝚚⟩⎮

|𝑎·𝚙+𝑏·𝚚| = √((𝑎·𝚙)²+(𝑏·𝚚)²) ≤ √(𝑎²+𝑏²)·√(𝚙²+𝚚²) = √(𝑎²𝚙²+𝑏²𝚙²+𝑎²𝚚²+𝑏²𝚚²)

𝑎²·𝚙²+𝑏²·𝚚² ≤ 𝑎²𝚙²+𝑏²𝚙²+𝑎²𝚚²+𝑏²𝚚²

0 ≤ 𝑎²𝚚²+𝑏²𝚙²

In general:

Equality holds if and only if either $(𝐯⃗)$ is a scalar multiple of $(𝐮⃗)$, or one of them is 0.

The dot product is never greater than the product of magnitudes. Only equal when direction aligns.