Vectors

To the vector belongs the spoils

Arrow Arithmetics

Vector:

• Magnitude
• Direction
• Sense

Example Signed numbers

  -3 -2 -1  0  1  2  3
┄┄─┼──┼──┼──┼──┼──┼──┼┄┄
   ⇆                 ⇆

Vectores

A⃗ = {aᵢ ∈ A} : 𝕂ₙ

Vector unitario

𝐴̂= A⃗╱|A⃗|

Dot product

A⃗·B⃗ 
	= ∑ aᵢ·bᵢ
	= |A⃗||B⃗| cosϑ


A⃗⟂B⃗ 
∴ A⃗·B⃗ = 0

Componente de A⃗ en dirección u᷍ , un Vector Unitario

	= |A⃗||u᷍| cosϑ
	= |A⃗|cosϑ
	= A⃗ · u᷍

Determinante

Det(A⃗, B⃗) =
⎪ 𝐚₁ 𝐚₂⎥
⎪      ⎜
⎪ 𝐛₁ 𝐛₂⎪
= 𝐚₁• 𝐛₂  -   𝐚₂• 𝐛₁ 
= |A⃗|•|B⃗|•sinθ
= Area of the Paralelogram 
 A⃗⏥
  B⃗

Determinante Espacial ℝ³

Volume

Det(A⃗, B⃗, 𝐶⃗) =
⎪ 𝐚₁ 𝐚₂ 𝐚₃⎥
⎪ 𝐛₁ 𝐛₂ 𝐛₃⎜ 
⎪ 𝒄₁ 𝒄₂ 𝒄₃⎪
===========
𝐚₁⎪ 𝐛₂ 𝐛₃⎜ 
  ⎪ 𝒄₂ 𝒄₃⎪
+
-𝐚₂⎪ 𝐛₁ 𝐛₃⎜ 
   ⎪ 𝒄₁ 𝒄₃⎪
+
𝐚₃⎪ 𝐛₁ 𝐛₂⎜ 
  ⎪ 𝒄₁ 𝒄₂⎪

Cross Product

Cross product is a binary operation on two vectors in three- dimensional space. It results in a vector that is perpendicular to both vectors. The Vector product of two vectors, a and b, is denoted by a × b. Its resultant vector is perpendicular to a and b. Vector products are also called cross products. Cross product of two vectors will give the resultant a vector and calculated using the Right-hand Rule.

A⃗×B⃗ 
= 
⎪ 𝕚᷍   𝕛᷍  𝕜᷍ ⎥
⎪ 𝐚₁ 𝐚₂ 𝐚₃⎥
⎪ 𝐛₁ 𝐛₂ 𝐛₃⎜ 

=
𝕚᷍ ⎪ 𝐚₂ 𝐚₃⎥
 ⎪ 𝐛₂ 𝐛₃⎜ 
+
𝕛᷍ ⎪ 𝐚₁ 𝐚₃⎥
 ⎪ 𝐛₁ 𝐛₃⎜ 
+
𝕜᷍
⎪ 𝐚₁ 𝐚₂⎥
⎪ 𝐛₁ 𝐛₂⎜ 

|A⃗×B⃗|
= Area of the Paralelogram 
 A⃗⏥
  B⃗

Dirección(A⃗×B⃗)
⏊ al plano A⃗,B⃗
Right Hand Rule

A⃗×B⃗=-B⃗×A⃗

Volumen

= area(base) • altura
= |A⃗×B⃗|(A⃗•𝒏⃗᷍)

Vectors

What is a vector?

• Arrow (?)

• Magnitude

• Direction

Magnitude

Measured
📏📐🔭
The way we measure a vector is context-dependent.

Direction

2D 2 numbers

float x;
float y;
[…]
x = x + xspeed;
y = y + yspeed;

Is the same as

PVector position;
…
position= position + speed