commit 281d9e2b08051abddccd58740f9a6f7f28fd13b6
parent 07d4f0d33d851db6ef9768a5ea55fab8c145ddc3
Author: ling0x <ling0x@users.noreply.github.com>
Date: Sun, 14 Jun 2026 12:15:25 +0100
chore: format
Diffstat:
8 files changed, 435 insertions(+), 598 deletions(-)
diff --git a/linear_algebra/geometry/links.md b/linear_algebra/geometry/links.md
@@ -1,13 +0,0 @@
-# Tesseract
-
-https://en.wikipedia.org/wiki/Tesseract
-
-# Geometric Continuity
-
-C1, G1 Surfaces, etc.:
-
-https://en.wikipedia.org/wiki/Smoothness#Geometric_continuity
-
-# Mobius Strip
-
-https://en.wikipedia.org/wiki/M%C3%B6bius_strip
diff --git a/linear_algebra/geometry/links.txt b/linear_algebra/geometry/links.txt
@@ -0,0 +1,13 @@
+===============================================================================
+GEOMETRY LINKS
+===============================================================================
+
+TESSERACT
+https://en.wikipedia.org/wiki/Tesseract
+
+GEOMETRIC CONTINUITY
+C1, G1 Surfaces, etc.:
+https://en.wikipedia.org/wiki/Smoothness#Geometric_continuity
+
+MOBIUS STRIP
+https://en.wikipedia.org/wiki/M%C3%B6bius_strip
diff --git a/linear_algebra/vectors/cross_product.md b/linear_algebra/vectors/cross_product.md
@@ -1,305 +0,0 @@
-# Cross Product
-
-The **cross product** is an operation that takes two vectors in
-$$ \mathbb{R}^3 $$
-
-and returns another vector in $$ \mathbb{R}^3 $$
-
-, written $\mathbf{a} \times \mathbf{b}$. Unlike the dot product, the result is
-a vector, not a scalar. The cross product is only defined in three dimensions
-(and in a generalized sense in seven dimensions; here we restrict to
-$\mathbb{R}^3$).
-
-## Geometric meaning
-
-- **Direction:** $\mathbf{a} \times \mathbf{b}$ is perpendicular to both
- $\mathbf{a}$ and $\mathbf{b}$, following the right-hand rule: if you point
- your fingers along $\mathbf{a}$ and curl them toward $\mathbf{b}$, your thumb
- points in the direction of $\mathbf{a} \times \mathbf{b}$.
-- **Magnitude:** $\|\mathbf{a} \times \mathbf{b}\| =
- \|\mathbf{a}\|\,\|\mathbf{b}\|\sin\theta$, where $\theta$ is the angle between
- $\mathbf{a}$ and $\mathbf{b}$. So the length equals the area of the
- parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$.
-
-## Algebraic definition
-
-For vectors
-
-$$
-\mathbf{a} =
-\begin{pmatrix}
-a_1\\
-a_2\\
-a_3
-\end{pmatrix},
-\quad
-\mathbf{b} =
-\begin{pmatrix}
-b_1\\
-b_2\\
-b_3
-\end{pmatrix},
-$$
-
-the cross product is
-
-$$
-\mathbf{a} \times \mathbf{b} =
-\begin{pmatrix}
-a_2 b_3 - a_3 b_2\\
-a_3 b_1 - a_1 b_3\\
-a_1 b_2 - a_2 b_1
-\end{pmatrix}.
-$$
-
-This can be remembered using the determinant of a formal $3\times 3$ matrix:
-
-$$
-
-\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 &
-\mathbf{e}_3\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix}
-
-$$
-
-$$
-
-\mathbf{e}_1(a_2 b_3 - a_3 b_2)
-
-- \mathbf{e}_2(a_1 b_3 - a_3 b_1)
-
-* \mathbf{e}_3(a_1 b_2 - a_2 b_1),
-
-$$
-
-where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard unit vectors
-in $\mathbb{R}^3$.
-
-### The "Cross-Out" Method (Fastest)
-
-The shorthand calculation for this is:
-
-1. Stack them: Write the components of the first vector over the second vector
- twice.
-2. Cross out the first and last columns.
-3. Multiply in an 'X' pattern (top-left bot-right minus top-right bot-left) for
- each remaining pair:
-
-<img src="/assets/cross_product_shorthand.png" alt="Cross Product Calculation" width="500">
-
-## Rules of calculation (with examples in LaTeX)
-
-Let $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^3$ and $\lambda \in
-\mathbb{R}$.
-
----
-
-**1. Anticommutativity**
-
-Swapping the order flips the sign:
-
-$$
-\mathbf{a} \times \mathbf{b} = -\bigl(\mathbf{b} \times \mathbf{a}\bigr).
-$$
-
-Example:
-
-$$
-\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} \times \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}
-= \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix},
-\quad
-\begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix} \times \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}
-= \begin{pmatrix} 0\\ 0\\ -1 \end{pmatrix}.
-$$
-
----
-
-**2. Distributivity over addition**
-
-$$
-\mathbf{a} \times (\mathbf{b} + \mathbf{c})
-= \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c},
-\qquad
-(\mathbf{a} + \mathbf{b}) \times \mathbf{c}
-= \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}.
-$$
-
-Example (second component of $\mathbf{a} \times (\mathbf{b}+\mathbf{c})$):
-
-$$
-\mathbf{a} = \begin{pmatrix} 1\\ 2\\ 0 \end{pmatrix},\;
-\mathbf{b} = \begin{pmatrix} 0\\ 1\\ 1 \end{pmatrix},\;
-\mathbf{c} = \begin{pmatrix} 1\\ 0\\ 1 \end{pmatrix}
-\;\Rightarrow\;
-\mathbf{b}+\mathbf{c} = \begin{pmatrix} 1\\ 1\\ 2 \end{pmatrix}.
-$$
-
-$$
-\mathbf{a} \times \mathbf{b} = \begin{pmatrix} 2\\ -1\\ 1 \end{pmatrix},\quad
-\mathbf{a} \times \mathbf{c} = \begin{pmatrix} 2\\ -1\\ -2 \end{pmatrix}
-\;\Rightarrow\;
-\mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} = \begin{pmatrix} 4\\ -2\\ -1 \end{pmatrix}.
-$$
-
-$$
-\mathbf{a} \times (\mathbf{b}+\mathbf{c}) = \begin{pmatrix} 2\cdot 2 - 0\cdot 1\\ 0\cdot 1 - 1\cdot 2\\ 1\cdot 1 - 2\cdot 1 \end{pmatrix} = \begin{pmatrix} 4\\ -2\\ -1 \end{pmatrix}.
-$$
-
----
-
-**3. Scalar multiplication (homogeneity)**
-
-A scalar can be factored out of either slot:
-
-$$
-(\lambda \mathbf{a}) \times \mathbf{b}
-= \mathbf{a} \times (\lambda \mathbf{b})
-= \lambda (\mathbf{a} \times \mathbf{b}).
-$$
-
-Example: with $\mathbf{a} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}$,
-$\mathbf{b} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}$, $\lambda = 3$,
-
-$$
-(3\mathbf{a}) \times \mathbf{b}
-= \begin{pmatrix} 3\\ 0\\ 0 \end{pmatrix} \times \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}
-= \begin{pmatrix} 0\\ 0\\ 3 \end{pmatrix}
-= 3 \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}
-= 3(\mathbf{a} \times \mathbf{b}).
-$$
-
----
-
-**4. Cross product with the zero vector**
-
-$$
-\mathbf{a} \times \mathbf{0} = \mathbf{0} \times \mathbf{a} = \mathbf{0}.
-$$
-
----
-
-**5. Parallel vectors**
-
-$\mathbf{a}$ and $\mathbf{b}$ are parallel (or one is zero) if and only if
-
-$$
-\mathbf{a} \times \mathbf{b} = \mathbf{0}.
-$$
-
-Example: $\mathbf{a} = \begin{pmatrix} 2\\ 4\\ 6 \end{pmatrix}$, $\mathbf{b}
-= \begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} = \tfrac{1}{2}\mathbf{a}$, so
-
-$$
-\mathbf{a} \times \mathbf{b}
-= \begin{pmatrix} 4\cdot 3 - 6\cdot 2\\ 6\cdot 1 - 2\cdot 3\\ 2\cdot 2 - 4\cdot 1 \end{pmatrix}
-= \begin{pmatrix} 0\\ 0\\ 0 \end{pmatrix}.
-$$
-
----
-
-**6. Self-cross product**
-
-$$
-\mathbf{a} \times \mathbf{a} = \mathbf{0}.
-$$
-
-(Special case of the parallel-vectors rule.)
-
----
-
-**7. Jacobi identity**
-
-$$
-
-\mathbf{a} \times (\mathbf{b} \times \mathbf{c})
-
-- \mathbf{b} \times (\mathbf{c} \times \mathbf{a})
-- \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}.
-
-$$
-
----
-
-**8. Relation to dot product (vector triple product expansion)**
-
-$$
-\mathbf{a} \times (\mathbf{b} \times \mathbf{c})
-= (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}.
-$$
-
-Example: $\mathbf{a} = \mathbf{e}_1$, $\mathbf{b} = \mathbf{e}_2$,
-$\mathbf{c} = \mathbf{e}_3$:
-
-$$
-\mathbf{a} \cdot \mathbf{c} = 0,\quad \mathbf{a} \cdot \mathbf{b} = 0
-\;\Rightarrow\;
-\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 0\cdot \mathbf{b} - 0\cdot \mathbf{c} = \mathbf{0}.
-$$
-
-$$
-\mathbf{b} \times \mathbf{c} = \mathbf{e}_1
-\;\Rightarrow\;
-\mathbf{e}_1 \times \mathbf{e}_1 = \mathbf{0}.
-$$
-
----
-
-**9. Magnitude and angle**
-
-$$
-\|\mathbf{a} \times \mathbf{b}\|^2
-= \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 - (\mathbf{a} \cdot \mathbf{b})^2.
-$$
-
-Equivalently, $\|\mathbf{a} \times \mathbf{b}\| =
-\|\mathbf{a}\|\,\|\mathbf{b}\|\sin\theta$.
-
----
-
-**10. Relation to the dot product (scalar triple product)**
-
-$$
-\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})
-= \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})
-= \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}).
-$$
-
-This value is the (signed) volume of the parallelepiped spanned by $\mathbf{a},
-\mathbf{b}, \mathbf{c}$. Example:
-
-$$
-\mathbf{a} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix},\;
-\mathbf{b} = \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix},\;
-\mathbf{c} = \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}
-\;\Rightarrow\;
-\mathbf{b} \times \mathbf{c} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix},\quad
-\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1.
-$$
-
-## Worked example
-
-Compute $\mathbf{u} \times \mathbf{v}$ for
-
-$$
-\mathbf{u} = \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix},\qquad
-\mathbf{v} = \begin{pmatrix} 1\\ 4\\ -2 \end{pmatrix}.
-$$
-
-$$
-\mathbf{u} \times \mathbf{v}
-= \begin{pmatrix}
-(-1)(-2) - (3)(4)\\
-(3)(1) - (2)(-2)\\
-(2)(4) - (-1)(1)
-\end{pmatrix}
-= \begin{pmatrix}
-2 - 12\\
-3 + 4\\
-8 + 1
-\end{pmatrix}
-= \begin{pmatrix} -10\\ 7\\ 9 \end{pmatrix}.
-$$
-
-Check: $\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 2(-10) + (-1)(7) +
-3(9) = -20 - 7 + 27 = 0$, and $\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v})
-= 1(-10) + 4(7) + (-2)(9) = -10 + 28 - 18 = 0$, so the result is perpendicular
-to both $\mathbf{u}$ and $\mathbf{v}$.
diff --git a/linear_algebra/vectors/cross_product.txt b/linear_algebra/vectors/cross_product.txt
@@ -0,0 +1,251 @@
+===============================================================================
+CROSS PRODUCT
+===============================================================================
+
+The cross product is an operation that takes two vectors in R^3 and returns
+another vector in R^3, written a x b. Unlike the dot product, the result is a
+vector, not a scalar. The cross product is only defined in three dimensions
+(and in a generalized sense in seven dimensions; here we restrict to R^3).
+
+
+-------------------------------------------------------------------------------
+1. GEOMETRIC MEANING
+-------------------------------------------------------------------------------
+
+Direction:
+ a x b is perpendicular to both a and b, following the right-hand rule: if you
+ point your fingers along a and curl them toward b, your thumb points in the
+ direction of a x b.
+
+Magnitude:
+ ||a x b|| = ||a|| * ||b|| * sin(theta)
+
+ where theta is the angle between a and b. The length equals the area of the
+ parallelogram spanned by a and b.
+
+
+-------------------------------------------------------------------------------
+2. ALGEBRAIC DEFINITION
+-------------------------------------------------------------------------------
+
+For vectors:
+
+ | a1 | | b1 |
+ a = | a2 | b = | b2 |
+ | a3 | | b3 |
+
+the cross product is:
+
+ | a2*b3 - a3*b2 |
+ a x b = | a3*b1 - a1*b3 |
+ | a1*b2 - a2*b1 |
+
+This can be remembered using the determinant of a formal 3x3 matrix:
+
+ | e1 e2 e3 |
+ | a1 a2 a3 | = e1*(a2*b3 - a3*b2)
+ | b1 b2 b3 | - e2*(a1*b3 - a3*b1)
+ + e3*(a1*b2 - a2*b1)
+
+where e1, e2, e3 are the standard unit vectors in R^3.
+
+
+-------------------------------------------------------------------------------
+3. THE "CROSS-OUT" METHOD (FASTEST)
+-------------------------------------------------------------------------------
+
+1. Stack them: write the components of the first vector over the second twice.
+
+2. Cross out the first and last columns.
+
+3. Multiply in an X pattern (top-left * bottom-right minus top-right *
+ bottom-left) for each remaining pair:
+
+ a1 a2 a3 a1 a2 a3
+ \/ \/ \/
+ /\ /\ /\
+ b1 b2 b3 b1 b2 b3
+
+ or:
+
+ a1 b1
+ a2 b2
+ \/
+ /\
+ a3 b3
+ \/
+ /\
+ a1 b1
+ \/
+ /\
+ a2 b2
+ a3 b3
+
+Result:
+
+ | a2*b3 - a3*b2 |
+ a x b = | a3*b1 - a1*b3 |
+ | a1*b2 - a2*b1 |
+
+
+-------------------------------------------------------------------------------
+4. RULES OF CALCULATION (WITH EXAMPLES)
+-------------------------------------------------------------------------------
+
+Let a, b, c in R^3 and lambda in R.
+
+
+4.1 Anticommutativity
+
+Swapping the order flips the sign:
+
+ a x b = -(b x a)
+
+Example:
+
+ | 1 | | 0 | | 0 |
+ | 0 | x | 1 | = | 0 |
+ | 0 | | 0 | | 1 |
+
+ | 0 | | 1 | | 0 |
+ | 1 | x | 0 | = | 0 |
+ | 0 | | 0 | | -1 |
+
+
+4.2 Distributivity over addition
+
+ a x (b + c) = a x b + a x c
+ (a + b) x c = a x c + b x c
+
+Example (second component of a x (b + c)):
+
+ | 1 | | 0 | | 1 |
+ a = | 2 | b = | 1 | c = | 0 |
+ | 0 | | 1 | | 1 |
+
+ b + c = | 1 |
+ | 1 |
+ | 2 |
+
+ a x b = | 2 | a x c = | 2 |
+ | -1 | | -1 |
+ | 1 | | -2 |
+
+ a x b + a x c = | 4 |
+ | -2 |
+ | -1 |
+
+ a x (b + c) = | 2*2 - 0*1 | | 4 |
+ | 0*1 - 1*2 | = | -2 |
+ | 1*1 - 2*1 | | -1 |
+
+
+4.3 Scalar multiplication (homogeneity)
+
+A scalar can be factored out of either slot:
+
+ (lambda*a) x b = a x (lambda*b) = lambda * (a x b)
+
+Example: a = | 1 |, b = | 0 |, lambda = 3
+
+ (3*a) x b = | 3 | | 0 | | 0 |
+ | 0 | x | 1 | = | 0 |
+ | 0 | | 0 | | 3 |
+
+ = 3 * | 0 | = 3 * (a x b)
+ | 0 |
+ | 1 |
+
+
+4.4 Cross product with the zero vector
+
+ a x 0 = 0 x a = 0
+
+
+4.5 Parallel vectors
+
+a and b are parallel (or one is zero) if and only if:
+
+ a x b = 0
+
+Example: a = | 2 |, b = | 1 | = (1/2)*a
+
+ | 4 | | 2 |
+ | 6 | | 3 |
+
+ a x b = | 4*3 - 6*2 | | 0 |
+ | 6*1 - 2*3 | = | 0 |
+ | 2*2 - 4*1 | | 0 |
+
+
+4.6 Self-cross product
+
+ a x a = 0
+
+(Special case of the parallel-vectors rule.)
+
+
+4.7 Jacobi identity
+
+ a x (b x c) - b x (c x a) - c x (a x b) = 0
+
+
+4.8 Relation to dot product (vector triple product expansion)
+
+ a x (b x c) = (a . c)*b - (a . b)*c
+
+Example: a = e1, b = e2, c = e3:
+
+ a . c = 0, a . b = 0
+ => a x (b x c) = 0*b - 0*c = 0
+
+ b x c = e1
+ => e1 x e1 = 0
+
+
+4.9 Magnitude and angle
+
+ ||a x b||^2 = ||a||^2 * ||b||^2 - (a . b)^2
+
+Equivalently:
+
+ ||a x b|| = ||a|| * ||b|| * sin(theta)
+
+
+4.10 Relation to the dot product (scalar triple product)
+
+ a . (b x c) = b . (c x a) = c . (a x b)
+
+This value is the (signed) volume of the parallelepiped spanned by a, b, c.
+
+Example:
+
+ a = | 1 | b = | 0 | c = | 0 |
+ | 0 | | 1 | | 0 |
+ | 0 | | 0 | | 1 |
+
+ b x c = | 1 |, a . (b x c) = 1
+ | 0 |
+ | 0 |
+
+
+-------------------------------------------------------------------------------
+5. WORKED EXAMPLE
+-------------------------------------------------------------------------------
+
+Compute u x v for:
+
+ | 2 | | 1 |
+ u = | -1 | v = | 4 |
+ | 3 | | -2 |
+
+ u x v = | (-1)*(-2) - 3*4 | | 2 - 12 | | -10 |
+ | 3*1 - 2*(-2) | = | 3 + 4 | = | 7 |
+ | 2*4 - (-1)*1 | | 8 + 1 | | 9 |
+
+Check:
+
+ u . (u x v) = 2*(-10) + (-1)*7 + 3*9 = -20 - 7 + 27 = 0
+ v . (u x v) = 1*(-10) + 4*7 + (-2)*9 = -10 + 28 - 18 = 0
+
+So the result is perpendicular to both u and v.
diff --git a/linear_algebra/vectors/dot_product.md b/linear_algebra/vectors/dot_product.md
@@ -1,231 +0,0 @@
-# Dot Product
-
-The dot product is an operation that takes two vectors of the same dimension and
-returns a single real number (a scalar), often written with a centered dot like
-\(\mathbf{a} \cdot \mathbf{b}\).
-
-## Algebraic definition
-
-For vectors in $$ \mathbb{R}^n $$
-
-$$
-\mathbf{a} =
-\begin{pmatrix}
-a_1\\
-a_2\\
-\vdots\\
-a_n
-\end{pmatrix},
-\quad
-\mathbf{b} =
-\begin{pmatrix}
-b_1\\
-b_2\\
-\vdots\\
-b_n
-\end{pmatrix},
-$$
-
-their dot product is
-
-$$
-
-\mathbf{a} \cdot \mathbf{b} =
-
-a_1 b_1 + a_2 b_2 + \dots + a_n b_n.
-
-$$
-
-Example in $$ \mathbb{R}^3 $$
-
-$$
-
-\begin{pmatrix} 1\\ 3\\ -5 \end{pmatrix} \cdot \begin{pmatrix} 4\\ -2\\ -1
-\end{pmatrix} =
-
-1\cdot 4 + 3\cdot(-2) + (-5)\cdot(-1) =
-
-4 - 6 + 5 = 3.
-
-$$
-
-## Geometric definition
-
-If $$ \mathbf{a}, \mathbf{b} \in \mathbb{R}^n $$
-
-and $$ \theta $$
-
-is the angle between them, then
-
-$$ \mathbf{a} \cdot \mathbf{b} $$
-
-$$ \|\mathbf{a}\|\;\|\mathbf{b}\|\cos\theta, $$
-
-where $$ \|\mathbf{a}\| $$
-
-is the Euclidean length (norm) of $$ \mathbf{a} $$
-
-From this, you also get
-
-$$
-\mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2,
-\quad
-\|\mathbf{a}\| = \sqrt{\mathbf{a} \cdot \mathbf{a}}.
-$$
-
-## Basic calculation rules
-
-Let $$ \mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^n $$
-
-and $$ \lambda \in
-\mathbb{R} $$
-
-Then:
-
-- Commutativity:
-
- $$
- \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.
- $$
-
-- Distributivity over addition:
-
- $$
- \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) =
- \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.
- $$
-
-- Homogeneity (scalar multiplication in one slot):
-
- $$
- (\lambda \mathbf{a}) \cdot \mathbf{b} =
- \lambda (\mathbf{a} \cdot \mathbf{b}), \quad \mathbf{a} \cdot (\lambda \mathbf{b}) =
- \lambda (\mathbf{a} \cdot \mathbf{b}).
- $$
-
-- Positivity:
-
- $$
- \mathbf{a} \cdot \mathbf{a} \ge 0
- \quad \text{and} \quad
- \mathbf{a} \cdot \mathbf{a} = 0 \iff \mathbf{a} = \mathbf{0}.
- $$
-
-## Worked examples
-
-1. Simple 2D example
-
-Let
-
-$$
-\mathbf{u} =
-\begin{pmatrix}
-2\\
--1
-\end{pmatrix},
-\quad
-\mathbf{v} =
-\begin{pmatrix}
-3\\
-4
-\end{pmatrix}.
-$$
-
-Then
-
-$$ \mathbf{u} \cdot \mathbf{v}
-2\cdot 3 + (-1)\cdot 4 =
-6 - 4 =
-2.
-$$
-
-2. 4D example
-
-Let
-
-$$
-\mathbf{x} =
-\begin{pmatrix}
-2\\
-0\\
--3\\
-1
-\end{pmatrix},
-\quad
-\mathbf{y} =
-\begin{pmatrix}
--1\\
-3\\
-1\\
-2
-\end{pmatrix}.
-$$
-
-Then
-
-$$
-\mathbf{x} \cdot \mathbf{y} =
-2(-1) + 0(3) + (-3)(1) + 1(2) =
--2 + 0 - 3 + 2 =
--3.
-$$
-
-3. Using the geometric form to find an angle
-
-Let
-
-$$
-\mathbf{a} =
-\begin{pmatrix}
-1\\
-2
-\end{pmatrix},
-\quad
-\mathbf{b} =
-\begin{pmatrix}
-2\\
-1
-\end{pmatrix}.
-$$
-
-Compute
-
-$$
-
-\mathbf{a} \cdot \mathbf{b} = 1\cdot 2 + 2\cdot 1 4,
-
-$$
-
-$$
-
-\|\mathbf{a}\| =
-
-\sqrt{1^2 + 2^2} =
-
-\sqrt{5}, \quad \|\mathbf{b}\| =
-
-\sqrt{2^2 + 1^2} =
-
-\sqrt{5}.
-
-$$
-
-So
-
-$$
-
-\cos\theta =
-
-\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\,\|\mathbf{b}\|} =
-
-\frac{4}{\sqrt{5}\sqrt{5}} =
-
-\frac{4}{5},
-
-$$
-
-and hence
-
-$$
-\theta = \arccos\!\left(\frac{4}{5}\right).
-$$
diff --git a/linear_algebra/vectors/dot_product.txt b/linear_algebra/vectors/dot_product.txt
@@ -0,0 +1,109 @@
+===============================================================================
+DOT PRODUCT
+===============================================================================
+
+The dot product is an operation that takes two vectors of the same dimension and
+returns a single real number (a scalar), often written as a . b.
+
+
+-------------------------------------------------------------------------------
+1. ALGEBRAIC DEFINITION
+-------------------------------------------------------------------------------
+
+For vectors in R^n:
+
+ | a1 | | b1 |
+ | a2 | | b2 |
+ a = | .. | b = | .. |
+ | an | | bn |
+
+their dot product is:
+
+ a . b = a1*b1 + a2*b2 + ... + an*bn
+
+Example in R^3:
+
+ | 1 | | 4 |
+ | 3 | . | -2 | = 1*4 + 3*(-2) + (-5)*(-1) = 4 - 6 + 5 = 3
+ | -5 | | -1 |
+
+
+-------------------------------------------------------------------------------
+2. GEOMETRIC DEFINITION
+-------------------------------------------------------------------------------
+
+If a, b in R^n and theta is the angle between them, then:
+
+ a . b = ||a|| * ||b|| * cos(theta)
+
+where ||a|| is the Euclidean length (norm) of a.
+
+From this, you also get:
+
+ a . a = ||a||^2
+ ||a|| = sqrt(a . a)
+
+
+-------------------------------------------------------------------------------
+3. BASIC CALCULATION RULES
+-------------------------------------------------------------------------------
+
+Let a, b, c in R^n and lambda in R. Then:
+
+Commutativity:
+
+ a . b = b . a
+
+Distributivity over addition:
+
+ a . (b + c) = a . b + a . c
+
+Homogeneity (scalar multiplication in one slot):
+
+ (lambda*a) . b = lambda * (a . b)
+ a . (lambda*b) = lambda * (a . b)
+
+Positivity:
+
+ a . a >= 0
+ a . a = 0 if and only if a = 0
+
+
+-------------------------------------------------------------------------------
+4. WORKED EXAMPLES
+-------------------------------------------------------------------------------
+
+4.1 Simple 2D example
+
+ | 2 |
+ u = | -1 | v = | 3 |
+ | 4 |
+
+ u . v = 2*3 + (-1)*4 = 6 - 4 = 2
+
+
+4.2 4D example
+
+ | 2 | | -1 |
+ | 0 | | 3 |
+ x = | -3 | y = | 1 |
+ | 1 | | 2 |
+
+ x . y = 2*(-1) + 0*3 + (-3)*1 + 1*2 = -2 + 0 - 3 + 2 = -3
+
+
+4.3 Using the geometric form to find an angle
+
+ | 1 | | 2 |
+ a = | 2 | b = | 1 |
+
+ a . b = 1*2 + 2*1 = 4
+
+ ||a|| = sqrt(1^2 + 2^2) = sqrt(5)
+ ||b|| = sqrt(2^2 + 1^2) = sqrt(5)
+
+ cos(theta) = (a . b) / (||a|| * ||b||)
+ = 4 / (sqrt(5) * sqrt(5))
+ = 4/5
+
+ theta = arccos(4/5)
diff --git a/linear_algebra/vectors/vectors.md b/linear_algebra/vectors/vectors.md
@@ -1,49 +0,0 @@
-# Vectors
-
-A vector is an object that has both **magnitude** (length) and **direction** and
-is often represented as an ordered list of numbers, like components along
-coordinate axes. In $\mathbb{R}^n$, a vector is typically written as a column or
-row of $n$ real numbers and can be added to other vectors or scaled by real
-numbers.
-
-1. Basic vector notation (inline):
-
-A vector in 2D can be written as $\vec{v} = (v_1, v_2)$.
-
-2. Column vector (display):
-
-A column vector in 3D:
-
-$$
-\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}
-$$
-
-3. Vector in $\mathbb{R}^n$:
-
-In general, a vector in $\mathbb{R}^n$ is
-
-$$
-\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \dots \\ v_n \end{bmatrix}.
-$$
-
-4. Vector addition and scalar multiplication:
-
-If $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$, then
-
-$$
-\vec{u} + \vec{v} = (u_1 + v_1,\; u_2 + v_2)
-$$
-
-and for a scalar $a$,
-
-$$
-a\vec{v} = (av_1,\; av_2).
-$$
-
-5. Magnitude (length) of a vector:
-
-The length of $\vec{v} = (v_1, v_2, v_3)$ is
-
-$$
-\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}.
-$$
diff --git a/linear_algebra/vectors/vectors.txt b/linear_algebra/vectors/vectors.txt
@@ -0,0 +1,62 @@
+===============================================================================
+VECTORS
+===============================================================================
+
+A vector is an object that has both magnitude (length) and direction and is
+often represented as an ordered list of numbers, like components along
+coordinate axes. In R^n, a vector is typically written as a column or row of n
+real numbers and can be added to other vectors or scaled by real numbers.
+
+
+-------------------------------------------------------------------------------
+1. BASIC VECTOR NOTATION (INLINE)
+-------------------------------------------------------------------------------
+
+A vector in 2D can be written as:
+
+ v = (v1, v2)
+
+
+-------------------------------------------------------------------------------
+2. COLUMN VECTOR (DISPLAY)
+-------------------------------------------------------------------------------
+
+A column vector in 3D:
+
+ | v1 |
+ v = | v2 |
+ | v3 |
+
+
+-------------------------------------------------------------------------------
+3. VECTOR IN R^n
+-------------------------------------------------------------------------------
+
+In general, a vector in R^n is:
+
+ | v1 |
+ | v2 |
+ v = | .. |
+ | vn |
+
+
+-------------------------------------------------------------------------------
+4. VECTOR ADDITION AND SCALAR MULTIPLICATION
+-------------------------------------------------------------------------------
+
+If u = (u1, u2) and v = (v1, v2), then:
+
+ u + v = (u1 + v1, u2 + v2)
+
+and for a scalar a:
+
+ a*v = (a*v1, a*v2)
+
+
+-------------------------------------------------------------------------------
+5. MAGNITUDE (LENGTH) OF A VECTOR
+-------------------------------------------------------------------------------
+
+The length of v = (v1, v2, v3) is:
+
+ ||v|| = sqrt(v1^2 + v2^2 + v3^2)
